(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose [itex]g(x) = \int_0^x f(t) dt[/itex], where [itex]f[/itex] is Lebesgue integrable on [itex]\mathbb R[/itex]. Give an [itex]\epsilon - \delta[/itex] proof that [itex]g'(y) = f(y)[/itex] if [itex]y\in (0,\infty)[/itex] is a point of continuity of [itex]f[/itex].

2. Relevant equations

3. The attempt at a solution

I know I need to show that

[tex]

f(y) = \lim_{h\to 0} \int_y^{y+h} \frac{1}{h} f(t) dt.

[/tex]

My idea was to try to do this in terms of sequences; i.e., to let [itex]\{h_n\}[/itex] be any sequence of real numbers such that [itex]h_n \to 0[/itex], and then to phrase the limit above in terms of a limit as [itex]n\to \infty[/itex]. I had then planned to use something like the dominated convergence theorem. But I don't have any idea how to make use of the hypothesis that [itex]f[/itex] is continuous at [itex]y[/itex], so I'm not sure if this is the right approach.

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# Homework Help: Differentiation and Lebesgue integration

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