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The Search for Primitive Functions. The Existence of Derivatives

In: Lebesgue’s Theory of Integration

Author

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  • Rahul Jain

    (Indian Institute of Science (IISc), Indian Police Service
    Tata Institute of Fundamental Research (TIFR))

Abstract

Let $$\mathfrak {F}(x)$$ 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 $$r[\mathfrak {F}(x), x, x+h]$$ r [ F ( x ) , x , x + h ] is also bounded, for any x and h. Since f(x) is the limit of $$r[\mathfrak {F}(x), x, x+h]$$ r [ F ( x ) , x , x + h ] for $$h = 0$$ h = 0 we can write, according to a theorem stated in Sect. 8.4.

Suggested Citation

  • Rahul Jain, 2025. "The Search for Primitive Functions. The Existence of Derivatives," Springer Books, in: Lebesgue’s Theory of Integration, chapter 0, pages 187-208, Springer.
    • Handle: RePEc:spr:sprchp:978-981-96-1169-0_10
    DOI: 10.1007/978-981-96-1169-0_10
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