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處處不連續函數 Fonction continue nulle part Nowhere continuous function
rdfs:comment
處處不連續函數是一數學名詞，是指在其定義域上的每一點都不連續的函數。若f(x)為一函數，定義域和值域都是實數，若針對每一個x，都存在ε > 0 ，使得針對每一個δ > 0，都可以找到y，使下式成立，則f(x)為處處不連續函數： 0< |x − y| < δ 且|f(x) − f(y)| ≥ ε 換句話說，不論距固定點多近，都有距固定點更近的點使函數的值偏離固定點對應的值。例如狄利克雷函数就是一個處處不連續函數。 若將定義中的絕對值改為度量空间中的距離或是拓扑空间中的類似名詞．即可定義更泛用的處處不連續函數。 En mathématiques, une fonction nulle part continue, également appelée fonction discontinue partout, est une fonction qui n'est continue en aucun point de son domaine. Si f est une fonction définie sur les nombres réels à valeur dans les nombres réels, alors f est nulle part continue si pour chaque point x il existe un ε > 0 tel que pour chaque δ > 0 nous pouvons trouver un point y tel que et . Par conséquent, peu importe à quel point nous nous rapprochons d'un point fixé, il existe des points encore plus proches auxquels la fonction prend des valeurs qui ne sont pas proches. In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers to real numbers, then f is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that 0 < |x − y| < δ and |f(x) − f(y)| ≥ ε. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
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Dirichlet-function
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dbo:abstract
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers to real numbers, then f is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that 0 < |x − y| < δ and |f(x) − f(y)| ≥ ε. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space. 處處不連續函數是一數學名詞，是指在其定義域上的每一點都不連續的函數。若f(x)為一函數，定義域和值域都是實數，若針對每一個x，都存在ε > 0 ，使得針對每一個δ > 0，都可以找到y，使下式成立，則f(x)為處處不連續函數： 0< |x − y| < δ 且|f(x) − f(y)| ≥ ε 換句話說，不論距固定點多近，都有距固定點更近的點使函數的值偏離固定點對應的值。例如狄利克雷函数就是一個處處不連續函數。 若將定義中的絕對值改為度量空间中的距離或是拓扑空间中的類似名詞．即可定義更泛用的處處不連續函數。 En mathématiques, une fonction nulle part continue, également appelée fonction discontinue partout, est une fonction qui n'est continue en aucun point de son domaine. Si f est une fonction définie sur les nombres réels à valeur dans les nombres réels, alors f est nulle part continue si pour chaque point x il existe un ε > 0 tel que pour chaque δ > 0 nous pouvons trouver un point y tel que et . Par conséquent, peu importe à quel point nous nous rapprochons d'un point fixé, il existe des points encore plus proches auxquels la fonction prend des valeurs qui ne sont pas proches. Des définitions plus générales de ce type de fonction peuvent être obtenues, en remplaçant la valeur absolue par la fonction distance dans un espace métrique, ou en utilisant la définition de continuité dans un espace topologique .
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