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Let \(\mathfrak {F}(x)\) be a function having a derivative f(x), we know that f(x) is measurable because it is a function of first class (Sect. 7.2). Let us assume that f(x) is bounded, then \(r[\mathfrak {F}(x), x, x+h]\) is also bounded, for any x and h. Since f(x) is the limit of \(r[\mathfrak {F}(x), x, x+h]\) for \(h = 0\) we can write, according to a theorem stated in Sect. 8.4.

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The Search for Primitive Functions. The Existence of Derivatives

  • 章节

  • 发表日期：

    2025年

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  • Rahul Jain

摘要

Let \(\mathfrak {F}(x)\) be a function having a derivative f(x), we know that f(x) is measurable because it is a function of first class (Sect. 7.2). Let us assume that f(x) is bounded, then \(r[\mathfrak {F}(x), x, x+h]\) is also bounded, for any x and h. Since f(x) is the limit of \(r[\mathfrak {F}(x), x, x+h]\) for \(h = 0\) we can write, according to a theorem stated in Sect. 8.4.