Function of bounded variation

Japanese: 有界変動関数 - ゆうかいへんどうかんすう（英語表記）function of bounded variation

For a function f ( x ) defined on the interval [ a , b ], if the sum created over the division ⊿ of the interval: a ₌ x0 < x1 < _... < xn = _b does not exceed a constant unrelated to the division, f is said to have bounded variation, and the upper limit V _of v⊿ when all divisions ⊿ are considered is called the total variation of f in [ a , b ]. Bounded variation functions are bounded. The linear combination and product of two bounded variation functions f and g are bounded variation, and furthermore, if there is c such that | g ( x ) | ≧ c ＞ 0, then f ( x )/ g ( x ) is also bounded variation.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

区間[a,b]で定義された関数f(x)に対して，区間の分割⊿：a＝x₀＜x₁＜……＜x_n＝bについて作った和，が，分割に無関係な定数を超えないとき，fは有界変動であるといい，あらゆる分割⊿を考えたときのv_⊿の上限Vを，fの[a,b]における全変動という。有界変動関数は有界である。二つの有界変動関数f,gの一次結合および積は有界変動であり，さらに｜g(x)｜≧c＞0なるcがあればf(x)/g(x)も有界変動である。

出典　株式会社平凡社世界大百科事典第２版について　情報

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