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In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in

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  • Bounded variation
  • Beschränkte Variation
  • Fonction à variation bornée
  • Funzione a variazione limitata
  • 有界変動函数
  • 有界变差
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  • In der Analysis ist eine Funktion von beschränkter Variation (beschränkter Schwankung), wenn ihre totale Variation (totale Schwankung) endlich ist, sie also in gewisser Weise nicht beliebig stark oszilliert. Diese Begriffe hängen eng mit der Stetigkeit und der Integrierbarkeit von Funktionen zusammen. Der Raum aller Funktionen von beschränkter Variation auf dem Gebiet wird mit bezeichnet.
  • En analyse, une fonction est dite à variation bornée quand elle vérifie une certaine condition de régularité. Cette condition a été introduite en 1881 par le mathématicien Camille Jordan pour étendre le théorème de Dirichlet sur la convergence des séries de Fourier.
  • 有界變差是函數的一個性質,它指的是總變差為有限的函數。 有界變差的理論對黎曼-斯蒂尔杰斯积分有相當的用處。
  • In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in
  • In analisi matematica, una branca della matematica, una funzione di variabile reale si dice a variazione limitata se la sua "variazione totale" è finita. Intuitivamente, le funzioni a variazione limitata in una variabile sono quelle per cui la distanza percorsa da un punto che si muove lungo il suo grafico è finita in ogni intervallo finito. Una funzione che non è a variazione limitata è il cosiddetto "seno del topologo", cioè se considerata in un qualsiasi intervallo che contenga lo 0, poiché all'avvicinarsi di a 0, la curva presenta infinite oscillazioni tra -1 e 1.
  • 解析学における有界変動の函数(ゆうかいへんどうのかんすう、英: fonction of bounded variation)あるいは有界変動函数(BV-function; BV函数)は、その変動が有界、すなわち全変動が有限値となるような実数値函数を言う。この性質は函数のグラフが以下に述べる意味において素性のよい (well behaved) ものであることを述べるものである。すなわち、一変数の連続函数が有界変動であるというのは、グラフ上を奔る動点の(x-軸方向への寄与分は無視して)y-軸方向への移動距離が有限であることを意味する。多変数の連続函数の場合にもこれは同様の意味を持つのであるが、考えるべき動点の辿る連続な路としては、与えられた函数のグラフ全体(今の場合これは超曲面になる)を取ることができないという事実があるので、グラフと固定された x-軸および y-軸に平行な任意の超平面との交叉を取る必要がある。 * 有界変動の函数があれば、その函数に関するリーマン–スティルチェス積分が任意の連続函数に対して定められる。 * 別な特徴付けとして、有界閉区間(コンパクト区間)上の有界変動函数は二つの有界単調増大函数の差として表される。 多変数の場合、開集合上定義された函数が有界変動となるのは、その函数の弱微分(超函数の意味での微分)がベクトル値の有限ラドン測度となるときである。
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