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Could anyone help me: Which statement is true and justify the answer.

A. The number of points at which a continuous function from the set of real numbers to the set of real numbers not differentiable is always countable.

B. There is a function from the set of real numbers to the set of real numbers which is continuous at every point but nowhere differentiable.

Thanks

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  • $\begingroup$ Indeed, point $B$ is correct. The Weierstrass function is everywhere continuous but nowhere differentiable.In fact, you can show, in some sense, that "almost every" everywhere continuous function is nowhere differentiable. $\endgroup$ – астон вілла олоф мэллбэрг Apr 8 '17 at 9:52
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Note that if statement $B$ is true then statement $A$ is false (Try exerting a little bit efforts to see this).

Statement $B$ is true thanks to Weierstrass.

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