Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 5, Problem 9

Problem

Let $f$ be a continuous real function on $\mathbb{R}^1$., of which it is known that $f'(x)$ exists for all $x \neq 0$ and that $f'(x) \rightarrow 3$ as $x \rightarrow 0$. Does it follow that $f'(0)$ exists?

Answer

$f'(0) = \lim_{h \rightarrow 0}\frac{f(0+h)-f(0)}{h} = \lim_{h \rightarrow 0}\frac{\frac{d}{dh}(f(0+h)-f(0))}{\frac{d}{dh}h}=\lim_{h \rightarrow 0} f'(h) = 3$. 

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