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Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & x=0.\end{cases}$$ Please give me a hint.

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Add a constant. – Peter Franek Jun 29 '14 at 11:05
Actually that class of functions is often used for examples of differentiable functions that are not $C^{1}$: do you mean that? – Dario Jun 29 '14 at 11:26
A similar question. – Lucian Jun 29 '14 at 13:14

$x\mapsto x|x|$ should be a good candidate.

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Consider $F$ an antiderivative of $x\to |x|$

Is $F$ twice differentiable?

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Take the anti-derivative of the Weierstrass function which is continuous everywhere and differentiable nowhere.

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