TITLE: Are Continuous Functions Always Differentiable?
QUESTION [23 upvotes]: Are continuous functions always differentiable? Are there any examples in dimension $n > 1$?

REPLY [2 votes]: Continuity requires
$$\lim_{h\to0}f(x+h)-f(x)=0.$$ Differentiability is stronger:
$$\lim_{h\to0}\frac{f(x+h)-f(x)}h$$ must exist.
Hence you can find counter-examples of the form
$$f(x)=x g(x)$$ where $$\lim_{x\to0}g(x)$$ does not exist.
E.g. $g(x)=\text{sgn}(x)$ or $g(x)=\sin\dfrac1x$ or $g(x)=|x|^{-1/2}$.