Why does the uniformly continuity and existence of right derivative implies the differentiabilty?

Asked Aug 19 '17 at 12:20

Active Aug 19 '17 at 12:43

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Assume $f:\mathbb R\to \mathbb R$ is uniformly continuous function and has a right derivative $f'_+:\mathbb R\to \mathbb R$ which is also uniformly continuous. Where $f'_+$ is define by $$f'_+(x) = \lim_{h\to 0^+}\frac{f(x+h)-f(x)}{h}$$

Prove or disprove that $f$ is differentiable.

edited Aug 19 '17 at 12:43

jimjim

9,855

asked Aug 19 '17 at 12:20

Guy Fsone

25,237

f' is uniformly continous in all $\mathbb{R}$? – ictibones Aug 19 '17 at 12:32
yes on all $\mathbb R$ – Guy Fsone Aug 19 '17 at 12:33
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What have you tried (and seen in class/your textbook, that you can use) before? – Clement C. Aug 19 '17 at 12:46
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Also, relevant: https://math.stackexchange.com/questions/418737/continuous-right-derivative-implies-differentiability – Clement C. Aug 19 '17 at 12:51
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Hint: prove Rolle's theorem with $f_+^\prime$, then the mean value theorem, then your result. – Gribouillis Aug 19 '17 at 12:53

Why does the uniformly continuity and existence of right derivative implies the differentiabilty?

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