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rdfs:label
Funtzio jarraitu Kontinuerlig funktion Stetige Funktion Continue functie (analyse) Funzione continua دالة مستمرة Continuous function Función continua Função contínua Fonction continue Funció contínua Συνέχεια συνάρτησης Непрерывное отображение 連続写像 Fungsi kontinu Kontinua funkcio Funkcja ciągła 连续函数 Неперервна функція 연속 함수 Spojité zobrazení
rdfs:comment
Inom matematiken är en kontinuerlig funktion en funktion som inte gör några plötsliga hopp och inte har några avbrott, så att nästan lika värden in garanterar nästan lika värden ut. För en reellvärd funktion f med ett reellt argument kan man precisera detta som att man för varje givet reellt tal x0 där funktionen är definierad och varje given noggrannhet kan vara säker på att f(x) approximerar f(x0) med minst denna noggrannhet för alla x som ligger tillräckligt nära x0. Dalam matematika, fungsi kontinu dalam adalah jenis fungsi yang perubahan secara kontinu (sinambung, tanpa terpotong) pada variabel fungsi mengakibatkan perubahan kontinu pada nilai keluaran fungsi. Hal ini mengartikan nilai fungsi tidak pernah mengalami perubahan yang mendadak/tiba-tiba. Gagasan intuitif kekontinuan mengilustrasikan fungsi kontinu sebagai fungsi yang grafiknya dapat digambar tanpa mengangkat kapur dari papan tulis. Secara lebih teknis, fungsi dikatakan kontinu jika perubahan kecil pada nilai fungsi dapat dipastikan cukup dengan membuat perubahan kecil pada variabelnya. Fungsi yang tidak kontinu dikatakan fungsi takkontinu atau fungsi diskontinu. Sampai pada abad ke-19, matematikawan sangat mengandalkan konsep kekontinuan yang intuitif. Hal ini berubah sejak definisi epsil Непреры́вное отображе́ние (непрерывная функция) — отображение из одного пространства в другое, при котором близкие точки области определения переходят в близкие точки области значений. Наиболее общее определение формулируется для отображений топологических пространств: непрерывным считается отображение, при котором прообраз всякого открытого множества открыт.Непрерывность отображений других типов пространств — метрических, нормированных и тому подобных пространств — является непосредственным следствием общего (топологического) определения, но формулируется с использованием структур, заданных в соответствующих пространствах — метрики, нормы и так далее. En matemáticas, una función continua es aquella para la cual, intuitivamente, para puntos cercanos del dominio se producen pequeñas variaciones en los valores de la función; aunque en rigor, en un espacio métrico como en variable real, significa que pequeñas variaciones de la función implican que deben estar cercanos los puntos. Si la función no es continua, se dice que es discontinua. Informalmente, una función continua de ℝ en ℝ es aquella cuya gráfica puede dibujarse sin levantar el lápiz del papel (más formalmente su grafo es un conjunto conexo). In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. "... será chamado de função contínua, se ... os valores numéricos da diferença diminuem arbitrariamente, conforme varie ... " Cauchy (1821) introduziu o conceito de função contínua, onde pequenas variações em x produzem pequenas variações em . Weierstrass (1874) reformulou a definição de Cauchy, onde a diferença será arbitrariamente pequena, se a diferença for suficientemente pequena. 位相空間論において函数や写像が連続(れんぞく、英: continuous)であるというのは、ある特定の意味で位相空間の間の位相的構造を保つある種の準同型となっていることを意味し、それ自体が位相空間論における興味の対象ともなる。数学の他の領域における各種の連続性の定義も、位相空間論における連続性の定義から導出することができる。連続性は、空間の位相が同相(位相同型)であることの基礎となる概念であり、特に全単射な連続写像が同相写像であるための必要十分条件は、その逆写像もまた連続となることである。 連続でない写像あるいは函数は、不連続であると言う。 連続性と近しい関係にある概念として、一様連続性、同程度連続性、作用素の有界性などがある。 位相空間の間の写像の連続性の概念は、それが距離空間の間の連続函数の場合のような明確な「距離」の概念を一般には持たない分、より抽象的である。位相空間というのは、集合 X とその上の位相(あるいは)と呼ばれる X の部分集合族で(距離空間における開球体全体の成す族の持つ性質を一般化するように)合併と交叉に関する特定の条件を満足するものを組にしたもので、位相空間においても与えられた点の近傍について考えることができる。位相に属する各集合は X の(その位相に関する)開部分集合と呼ばれる。 Matematikan, funtzio bat jarraitua dela esaten da aldagai askean izandako aldaketa txikiek funtzioaren balioan ere aldaketa txikiak eragiten dituztenean. Jarraituak ez diren funtzioak ez-jarraituak dira. Alderantzizko jarraitua duen funtzio jarraituari deritzo. 连续函数(英語:Continuous function)是指函数在数学上的属性为连续。直观上来说,连续的函数就是当输入值的变化足够小的时候,输出的变化也会随之足够小的函数。 如果输入值的某种微小的变化会产生输出值的一个突然的跳跃甚至无法定义,则这个函数被称为是不连续函数,或者说具有不连续性。非连续函数一定存在间断点。 举例来说,考虑描述一棵树的高度随时间而变化的函数,那么这个函数是连续的(除非树被砍断)。又例如,假设表示地球上某一点的空气温度,则这个函数也是连续的。事实上,古典物理学中有一句格言:“自然界中,一切都是连续的。”相比之下,如果表述在时间t的时候银行账户上的钱币金额,则这个函数无论在存钱或者取钱的时候都会有跳跃,因此函数是不连续的。 Spojité zobrazení je pojem z topologie a matematické analýzy.Je to takové zobrazení, které zobrazuje dostatečně blízké body blízko sebe. Tato vlastnost zobrazení se nazývá spojitost. Spojité zobrazení je zobecněním pojmu spojitá funkce na množinách čísel. In matematica, una funzione continua è una funzione che, intuitivamente, fa corrispondere ad elementi sufficientemente vicini del dominio elementi arbitrariamente vicini del codominio. Esistono diverse definizioni di continuità, corrispondenti ai contesti matematici in cui vengono utilizzate: la continuità di una funzione è uno dei concetti di base della topologia e dell'analisi matematica. La continuità di una funzione può essere definita anche in modo locale: in questo caso si parla di continuità in un punto del dominio. Una funzione continua è, per definizione, continua in ogni punto del proprio dominio. Una funzione che non è continua è detta discontinua, e i punti del dominio in cui non è continua sono detti punti di discontinuità. In der Mathematik ist eine stetige Abbildung oder stetige Funktion eine Funktion, bei der hinreichend kleine Änderungen des Arguments nur beliebig kleine Änderungen des Funktionswerts nach sich ziehen. Formalisieren kann man diese Eigenschaft mit der Vertauschbarkeit der Funktion mit Grenzwerten oder mit dem --Kriterium. Viele in der Praxis der reellen Analysis verwendete Funktionen sind stetig, insbesondere ist das für alle differenzierbaren Funktionen der Fall. Een continue functie is in de wiskunde een functie waarvan kleine veranderingen van een variabele resulteren in kleine veranderingen van de functiewaarde. Continue afbeeldingen zijn onderwerp van studie in bijvoorbeeld de analyse en de topologie. 위상수학과 해석학에서 연속 함수(連續函數, 문화어: 련속함수, 영어: continuous function)는 정의역의 점의 ‘작은 변화’에 대하여, 치역의 값 역시 작게 변화하는 함수이다. Funció contínua és un terme utilitzat en matemàtiques i, en particular, en topologia. Funkcja ciągła – funkcja, którą intuicyjnie można scharakteryzować jako: 1. * funkcję, w której mała zmiana argumentu powoduje małą zmianę wartości funkcji; inaczej mówiąc, dla argumentów leżących blisko siebie wartości funkcji też leżą blisko, 2. * funkcję rzeczywistą (określoną na zbiorze lub jego podprzedziale), której wykresem jest ciągła linia, tj. linia narysowana bez odrywania ołówka od papieru. Zbiór staje się przestrzenią topologiczną, gdy dla każdego jego elementu określimy rodzinę otoczeń tego elementu – podzbiorów Musi ona spełniać pewne warunki. symbole i to kwantyfikatory. Στα μαθηματικά, μία συνάρτηση λέγεται συνεχής όταν μια μικρή μεταβολή στο όρισμά της προκαλεί μικρή μόνο μεταβολή στην τιμή της. Για τις συναρτήσεις που ορίζονται στους πραγματικούς αριθμούς η γραφική παράσταση μιας συνεχούς συνάρτησης σε ένα διάστημα (και όχι σε μία ένωση διαστημάτων) μπορεί να σχεδιαστεί χωρίς να χρειαστεί να σηκώσουμε το μολύβι από το χαρτί. En matematiko, kontinua funkcio estas funkcio, kies valoro malmulte ŝanĝiĝas en okazo de malgranda ŝanĝo de la argumento. Se malgranda ŝanĝo de la argumento povas produkti rompan salton en valoro de la funkcio, la funkcio estas nekontinua. La ĉirkaŭteksto de ĉi tiu termino estas reelo-valoraj funkcioj sur la reela domajno aŭ sur topologia aŭ metrika spacoj escepte la kompleksajn nombrojn. Pri komplekso-valoraj funkcioj vidu artikolon kompleksa analitiko. La rimarkinda diferenco en maniero estas tiu ke en la reela domajno, la punktoj en la domajno kiuj estas punktoj de nekontinueco estas specialaĵoj. Sed en la kompleksa domajno tiaj punktoj estas kutime aparte forprenitaj el la domajno, do la funkcio kontinua en kompleksa domajno estas kontinua sur malkonektita partoj de reela domajno. الدالة المستمرة أو الدالة المتصلة أو الدالة المتصلة بانتظام (بالإنجليزية: Continuous function)‏ هي دالة رياضية تؤدي فيها تغييرات طفيفة في متغيّر الدالّة إلى تغييرات طفيفة في قيمتها. الدالة التي لا تحقّق هذه الخاصّة تدعى (دالة غير مستمرة) أو (دالة منفصلة). بشكل بديهي، فإنّ دالة ما هي مستمرّة إذا استطعنا أن نرسم رسمها البياني بدون رفع القلم عن الورقة، مع أنّ هذا التعريف ليس دقيقًا. يعتبر موضوع استمراريّة الدوال أحد المواضيع المبدئية والجوهريّة في الطوبولوجيا. في هذه الصفحة، سيكون الحديث عن دالة ذات مصادر وقيم حقيقيّة. Непере́рвна фу́нкція — в математичному аналізі це функція, у якій малим змінам аргумента відповідають малі зміни значення функції. Це означає, що графік неперервної функції не має стрибків, тобто може бути накреслений «не відриваючи олівець від паперу». Усі елементарні функції — неперервні на своїй області визначення. Неперервні функції трапляються набагато частіше, ніж диференційовні, множина всіх неперервних функцій замкнена відносно арифметичних операцій (за винятком ділення) і композиції та утворює чи не найважливіший клас функцій в аналізі.
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A real-valued function is continuous at if its natural extension to the hyperreals has the property that for all infinitesimal , is infinitesimal
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Continuous function Proof
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Dalam matematika, fungsi kontinu dalam adalah jenis fungsi yang perubahan secara kontinu (sinambung, tanpa terpotong) pada variabel fungsi mengakibatkan perubahan kontinu pada nilai keluaran fungsi. Hal ini mengartikan nilai fungsi tidak pernah mengalami perubahan yang mendadak/tiba-tiba. Gagasan intuitif kekontinuan mengilustrasikan fungsi kontinu sebagai fungsi yang grafiknya dapat digambar tanpa mengangkat kapur dari papan tulis. Secara lebih teknis, fungsi dikatakan kontinu jika perubahan kecil pada nilai fungsi dapat dipastikan cukup dengan membuat perubahan kecil pada variabelnya. Fungsi yang tidak kontinu dikatakan fungsi takkontinu atau fungsi diskontinu. Sampai pada abad ke-19, matematikawan sangat mengandalkan konsep kekontinuan yang intuitif. Hal ini berubah sejak definisi epsilon-delta dari limit diperkenalkan untuk memformalkan definisi kekontinuan. Kekontinuan adalah salah satu konsep inti dalam kalkulus dan analisis matematika, yang membahas fungsi dengan keluaran maupun variabelnya dapat berupa bilangan real atau kompleks. Konsep kekontinuan juga diperumum untuk fungsi antar ruang metrik dan antar ruang topologis. Fungsi jenis terakhir adalah fungsi kontinu yang paling umum, dan definisinya menjadi dasar ilmu topologi. Sebagai contoh, fungsi h(t) yang memerikan tinggi bunga yang sedang tumbuh pada waktu t dapat dianggap fungsi kontinu. Sebaliknya, jika fungsi M(t) melambangkan jumlah uang di sebuah rekening bank pada waktu t, nilai fungsi ini akan "melompat" ketika uang disimpan atau ditarik. Hal ini menyebabkan M(t) adalah fungsi diskontinu. En matemáticas, una función continua es aquella para la cual, intuitivamente, para puntos cercanos del dominio se producen pequeñas variaciones en los valores de la función; aunque en rigor, en un espacio métrico como en variable real, significa que pequeñas variaciones de la función implican que deben estar cercanos los puntos. Si la función no es continua, se dice que es discontinua. Informalmente, una función continua de ℝ en ℝ es aquella cuya gráfica puede dibujarse sin levantar el lápiz del papel (más formalmente su grafo es un conjunto conexo). La continuidad de funciones es uno de los conceptos básicos del análisis matemático y de la topología general. Este artículo describe principalmente la continuidad de funciones reales de una variable real. 位相空間論において函数や写像が連続(れんぞく、英: continuous)であるというのは、ある特定の意味で位相空間の間の位相的構造を保つある種の準同型となっていることを意味し、それ自体が位相空間論における興味の対象ともなる。数学の他の領域における各種の連続性の定義も、位相空間論における連続性の定義から導出することができる。連続性は、空間の位相が同相(位相同型)であることの基礎となる概念であり、特に全単射な連続写像が同相写像であるための必要十分条件は、その逆写像もまた連続となることである。 連続でない写像あるいは函数は、不連続であると言う。 連続性と近しい関係にある概念として、一様連続性、同程度連続性、作用素の有界性などがある。 位相空間の間の写像の連続性の概念は、それが距離空間の間の連続函数の場合のような明確な「距離」の概念を一般には持たない分、より抽象的である。位相空間というのは、集合 X とその上の位相(あるいは)と呼ばれる X の部分集合族で(距離空間における開球体全体の成す族の持つ性質を一般化するように)合併と交叉に関する特定の条件を満足するものを組にしたもので、位相空間においても与えられた点の近傍について考えることができる。位相に属する各集合は X の(その位相に関する)開部分集合と呼ばれる。 In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions and . The latter are the most general continuous functions, and their definition is the basis of topology. A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity. As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn. In matematica, una funzione continua è una funzione che, intuitivamente, fa corrispondere ad elementi sufficientemente vicini del dominio elementi arbitrariamente vicini del codominio. Esistono diverse definizioni di continuità, corrispondenti ai contesti matematici in cui vengono utilizzate: la continuità di una funzione è uno dei concetti di base della topologia e dell'analisi matematica. La continuità di una funzione può essere definita anche in modo locale: in questo caso si parla di continuità in un punto del dominio. Una funzione continua è, per definizione, continua in ogni punto del proprio dominio. Una funzione che non è continua è detta discontinua, e i punti del dominio in cui non è continua sono detti punti di discontinuità. Per esempio, la funzione che descrive l'altezza di un uomo rispetto alla sua età può essere vista come una funzione continua: in periodi brevi l'uomo cresce di poco. Al contrario, la funzione che rappresenta la quantità di denaro presente in un conto corrente nel tempo è una funzione discontinua, poiché prelievi e depositi le fanno fare salti da un valore all'altro. 连续函数(英語:Continuous function)是指函数在数学上的属性为连续。直观上来说,连续的函数就是当输入值的变化足够小的时候,输出的变化也会随之足够小的函数。 如果输入值的某种微小的变化会产生输出值的一个突然的跳跃甚至无法定义,则这个函数被称为是不连续函数,或者说具有不连续性。非连续函数一定存在间断点。 举例来说,考虑描述一棵树的高度随时间而变化的函数,那么这个函数是连续的(除非树被砍断)。又例如,假设表示地球上某一点的空气温度,则这个函数也是连续的。事实上,古典物理学中有一句格言:“自然界中,一切都是连续的。”相比之下,如果表述在时间t的时候银行账户上的钱币金额,则这个函数无论在存钱或者取钱的时候都会有跳跃,因此函数是不连续的。 Funkcja ciągła – funkcja, którą intuicyjnie można scharakteryzować jako: 1. * funkcję, w której mała zmiana argumentu powoduje małą zmianę wartości funkcji; inaczej mówiąc, dla argumentów leżących blisko siebie wartości funkcji też leżą blisko, 2. * funkcję rzeczywistą (określoną na zbiorze lub jego podprzedziale), której wykresem jest ciągła linia, tj. linia narysowana bez odrywania ołówka od papieru. Ciągłość funkcji jest jednym z podstawowych pojęć topologii, gdzie jest definiowana w sposób najbardziej ogólny, rozszerzając pojęcie ciągłości funkcji zmiennych rzeczywistych oraz funkcji w przestrzeniach metrycznych. To ujęcie jest jednocześnie bardzo proste i pozwala jednolicie potraktować przypadki nieskończoności (bardzo potrzebne przy pojęciu granicy funkcji i granicy ciągu): Zbiór staje się przestrzenią topologiczną, gdy dla każdego jego elementu określimy rodzinę otoczeń tego elementu – podzbiorów Musi ona spełniać pewne warunki. Najczęściej spotykamy się z takimi przestrzeniami topologicznymi, których topologia jest wyznaczona przez metrykę, czyli sposób określania odległości punktów tej przestrzeni. Wtedy za otoczenia punktu przyjmuje się kule o środku w tym punkcie i dodatnim promieniu. Standardowo przyjmuje się kule otwarte, ale użycie kul domkniętych prowadzi do tej samej topologii. Natomiast gdyby dopuścić kule domknięte o promieniu 0, otrzymalibyśmy tzw. topologię dyskretną, na ogół inną od wprowadzonej przez metrykę. Inaczej jest z nieskończonościami – tu nie określamy otoczeń przez metrykę: otoczeniami elementu w rozszerzonym zbiorze liczb rzeczywistych są przedziały dla dowolnych otoczeniami elementu są przedziały dla dowolnych (ciekawe są otoczenia dla ). Niech będą dane zbiory i zawarte w przestrzeniach topologicznych, i funkcja Definicja (topologiczna): Funkcja jest ciągła w punkcie jeśli symbole i to kwantyfikatory. Definicja: Funkcja jest ciągła jeśli jest ciągła w każdym punkcie swojej dziedziny, czyli w zbiorze Funkcja jest ciągła w zbiorze zawartym w jej dziedzinie jeśli jest ciągła w każdym punkcie tego zbioru. W dalszym ciągu podajemy bardziej tradycyjne ujęcia i pokazujemy, że definicje Cauchy’ego są równoważne powyższym. Inom matematiken är en kontinuerlig funktion en funktion som inte gör några plötsliga hopp och inte har några avbrott, så att nästan lika värden in garanterar nästan lika värden ut. För en reellvärd funktion f med ett reellt argument kan man precisera detta som att man för varje givet reellt tal x0 där funktionen är definierad och varje given noggrannhet kan vara säker på att f(x) approximerar f(x0) med minst denna noggrannhet för alla x som ligger tillräckligt nära x0. Begreppet kontinuitet är dock mycket använt inom olika delar av matematiken, även sådana där denna intuitiva förklaring inte så lätt låter sig omformuleras i en stringent definition. Topologi är den gren av matematiken som studerar kontinuerliga funktioner i dess mest generella betydelse. Där definieras en funktion mellan två topologiska rum som kontinuerlig, om varje urbild av en öppen mängd är öppen. Man kan visa att denna generella definition betyder samma sak som den vanliga definitionen för "vanliga" funktioner. In der Mathematik ist eine stetige Abbildung oder stetige Funktion eine Funktion, bei der hinreichend kleine Änderungen des Arguments nur beliebig kleine Änderungen des Funktionswerts nach sich ziehen. Formalisieren kann man diese Eigenschaft mit der Vertauschbarkeit der Funktion mit Grenzwerten oder mit dem --Kriterium. Anschaulich gesprochen ist eine reelle stetige Funktion dadurch gekennzeichnet, dass ihr Graph in einem kartesischen Koordinatensystem innerhalb ihres Definitionsbereiches eine zusammenhängende Kurve ist, der Graph also keine Sprünge macht und man ihn ohne Absetzen des Stiftes zeichnen kann. Viele in der Praxis der reellen Analysis verwendete Funktionen sind stetig, insbesondere ist das für alle differenzierbaren Funktionen der Fall. Für stetige Funktionen können eine Reihe nützlicher Eigenschaften bewiesen werden. Exemplarisch seien der Zwischenwertsatz, der Satz vom Minimum und Maximum und der Fundamentalsatz der Analysis genannt. Allgemeiner ist das Konzept der Stetigkeit von Abbildungen in der Mathematik vor allem in den Teilgebieten der Analysis und der Topologie von zentraler Bedeutung.Es ist möglich, Stetigkeit durch eine Bedingung zu charakterisieren, die nur Begriffe der Topologie benutzt. Somit kann der Begriff der Stetigkeit auch auf Funktionen zwischen topologischen Räumen ausgedehnt werden. Diese allgemeine Sichtweise erweist sich aus mathematischer Sicht als der „natürlichste“ Zugang zum Stetigkeitsbegriff: Stetige Funktionen sind diejenigen Funktionen zwischen topologischen Räumen, die mit deren Strukturen „verträglich“ sind. Stetige Funktionen spielen also in Topologie und Analysis eine ähnliche Rolle wie Homomorphismen in der Algebra. 위상수학과 해석학에서 연속 함수(連續函數, 문화어: 련속함수, 영어: continuous function)는 정의역의 점의 ‘작은 변화’에 대하여, 치역의 값 역시 작게 변화하는 함수이다. Een continue functie is in de wiskunde een functie waarvan kleine veranderingen van een variabele resulteren in kleine veranderingen van de functiewaarde. Continue afbeeldingen zijn onderwerp van studie in bijvoorbeeld de analyse en de topologie. Veel bekende functies op de reële getallen, zoals voor , de e-macht, de functies sinus en cosinus, zijn continu. Ook zijn de som, het verschil en het product van twee continue functies weer continu. Er bestaan verschillende definities van het begrip continuïteit. De bekendste is de epsilon-delta-definitie, die de bovenstaande populaire formulering precisieert. Непере́рвна фу́нкція — в математичному аналізі це функція, у якій малим змінам аргумента відповідають малі зміни значення функції. Це означає, що графік неперервної функції не має стрибків, тобто може бути накреслений «не відриваючи олівець від паперу». Усі елементарні функції — неперервні на своїй області визначення. Неперервні функції трапляються набагато частіше, ніж диференційовні, множина всіх неперервних функцій замкнена відносно арифметичних операцій (за винятком ділення) і композиції та утворює чи не найважливіший клас функцій в аналізі. Funció contínua és un terme utilitzat en matemàtiques i, en particular, en topologia. Непреры́вное отображе́ние (непрерывная функция) — отображение из одного пространства в другое, при котором близкие точки области определения переходят в близкие точки области значений. Наиболее общее определение формулируется для отображений топологических пространств: непрерывным считается отображение, при котором прообраз всякого открытого множества открыт.Непрерывность отображений других типов пространств — метрических, нормированных и тому подобных пространств — является непосредственным следствием общего (топологического) определения, но формулируется с использованием структур, заданных в соответствующих пространствах — метрики, нормы и так далее. В математическом анализе и комплексном анализе, где рассматриваются числовые функции и их обобщения на случай многомерных пространств, непрерывность функции вводится на языке пределов: такие определения непрерывности были исторически первыми и послужили основой для формирования общего понятия. Существование непрерывных отображений между пространствами, позволяет «переносить» свойства одного пространства в другое: например, непрерывный образ компактного пространства также является компактным. Непрерывное отображение, которое обладает обратным и также непрерывным отображением, называется гомеоморфизмом. Гомеоморфизм порождает на классе топологических пространств отношение эквивалентности; пространства, гомеоморфные друг другу, обладают одними и теми же топологическими свойствами, а сами свойства, которые сохраняются при гомеоморфизмах, называются топологическими инвариантами. Matematikan, funtzio bat jarraitua dela esaten da aldagai askean izandako aldaketa txikiek funtzioaren balioan ere aldaketa txikiak eragiten dituztenean. Jarraituak ez diren funtzioak ez-jarraituak dira. Alderantzizko jarraitua duen funtzio jarraituari deritzo. Funtzioen jarraitutasuna topologiaren oinarrizko kontzeptua da, era orokorrean lantzen da aurrerago. Artikulu honen sarrera aldagai errealetako funtzioetan ardazten da. Jarraitutasunaren beste kasu sendoago bat da. Gainera, artikulu honetan bi espazio metrikoren arteko funtzioen jarraitutasunaren kasu orokorragoaren definizioa lantzen da. , bereziki , kontsideratu daiteke. Beste jarraitutasun motak existitzen dira, baina ez dira lantzen artikulu honetan. Jarraitutasunaren definizio zehatza badago ere, intuitiboki funtzioa jarraitua da definizio-eremuko tarte batean bere grafikoa arkatza paperetik altxatu gabe marraz daitekeenean. Adibidez, izan bedi funtzioa, lore baten altuera deskribatzen duenta denboran. Funtzio hau jarraitua da. Aldiz, -k banku kontu batean dagoen diru kopurua adierazten badu denboran, funtzioak salto egiten du dirua sartu edo ateratzen denean; beraz funtzio ez-jarraitua da. "... será chamado de função contínua, se ... os valores numéricos da diferença diminuem arbitrariamente, conforme varie ... " Cauchy (1821) introduziu o conceito de função contínua, onde pequenas variações em x produzem pequenas variações em . Weierstrass (1874) reformulou a definição de Cauchy, onde a diferença será arbitrariamente pequena, se a diferença for suficientemente pequena. Posteriormente, com um tratamento mais rigoroso da matemática e a consequente evolução do pensamento matemático, as funções contínuas foram abstraídas para outros campos além da análise: álgebra linear, álgebra abstrata, física matemática, etc. Στα μαθηματικά, μία συνάρτηση λέγεται συνεχής όταν μια μικρή μεταβολή στο όρισμά της προκαλεί μικρή μόνο μεταβολή στην τιμή της. Για τις συναρτήσεις που ορίζονται στους πραγματικούς αριθμούς η γραφική παράσταση μιας συνεχούς συνάρτησης σε ένα διάστημα (και όχι σε μία ένωση διαστημάτων) μπορεί να σχεδιαστεί χωρίς να χρειαστεί να σηκώσουμε το μολύβι από το χαρτί. En matematiko, kontinua funkcio estas funkcio, kies valoro malmulte ŝanĝiĝas en okazo de malgranda ŝanĝo de la argumento. Se malgranda ŝanĝo de la argumento povas produkti rompan salton en valoro de la funkcio, la funkcio estas nekontinua. La ĉirkaŭteksto de ĉi tiu termino estas reelo-valoraj funkcioj sur la reela domajno aŭ sur topologia aŭ metrika spacoj escepte la kompleksajn nombrojn. Pri komplekso-valoraj funkcioj vidu artikolon kompleksa analitiko. La rimarkinda diferenco en maniero estas tiu ke en la reela domajno, la punktoj en la domajno kiuj estas punktoj de nekontinueco estas specialaĵoj. Sed en la kompleksa domajno tiaj punktoj estas kutime aparte forprenitaj el la domajno, do la funkcio kontinua en kompleksa domajno estas kontinua sur malkonektita partoj de reela domajno. Spojité zobrazení je pojem z topologie a matematické analýzy.Je to takové zobrazení, které zobrazuje dostatečně blízké body blízko sebe. Tato vlastnost zobrazení se nazývá spojitost. Spojité zobrazení je zobecněním pojmu spojitá funkce na množinách čísel. الدالة المستمرة أو الدالة المتصلة أو الدالة المتصلة بانتظام (بالإنجليزية: Continuous function)‏ هي دالة رياضية تؤدي فيها تغييرات طفيفة في متغيّر الدالّة إلى تغييرات طفيفة في قيمتها. الدالة التي لا تحقّق هذه الخاصّة تدعى (دالة غير مستمرة) أو (دالة منفصلة). بشكل بديهي، فإنّ دالة ما هي مستمرّة إذا استطعنا أن نرسم رسمها البياني بدون رفع القلم عن الورقة، مع أنّ هذا التعريف ليس دقيقًا. يعتبر موضوع استمراريّة الدوال أحد المواضيع المبدئية والجوهريّة في الطوبولوجيا. في هذه الصفحة، سيكون الحديث عن دالة ذات مصادر وقيم حقيقيّة. على سبيل المثال، إذا كانت الدالّة تمثّل ارتفاع زهرة ما في الزمن ، فإنّ هذه الدالة مستمرة. في الواقع، فهنالك قول مأثور في الفيزياء الكلاسيكية يقضي بأنّ كل شيء في الطبيعة استمراري (مستمر). وإذا فرضنا أنّ الدالة تمثّل ارتفاع رصيد حساب في البنك في الزمن ، فإنّ قيمة الدالة «تقفز» كلّما تم سحب بعض المال أو إدخاله إلى الحساب، لذا فإنّ هذه الدالّة غير مستمرّة.
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A function is continuous at if and only if it is sequentially continuous at that point.
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Subject Item
dbr:Surface_charge
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Random_search
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Transcendental_function
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Temporal_mean
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Two-dimensional_filter
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Left-continuous
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Left_continuous
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Wright_omega_function
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Spline_wavelet
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Vietoris–Begle_mapping_theorem
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Theta_vacuum
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Selection_theorem
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:Simple_path
dbo:wikiPageWikiLink
dbr:Continuous_function
Subject Item
dbr:E-d_definition
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Discontinuous_function
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Discontinuity_set
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuity_at_a_point
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuity_property
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuity_space
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_at_a_point
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_binary_relation
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_extension
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_fctn
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_map_(topology)
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_relation
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_space
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Continuous_variation
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Real-valued_continuous_functions
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Cts_fctn
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Sequential_continuity
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
dbr:Stepping_Stone_Theorem
dbo:wikiPageWikiLink
dbr:Continuous_function
dbo:wikiPageRedirects
dbr:Continuous_function
Subject Item
wikipedia-en:Continuous_function
foaf:primaryTopic
dbr:Continuous_function