Can we find a continuous function f: R→R such that f is not differentiable on (-δ, 0) for sufficiently small positive δ?
When I first think about it, I mistakenly think of the left differentiability of the function at x = 0, which is a different concept, and so come up with wrong examples such as
$$ f(x) = \left\{ \begin{array}{ll} \sin\Big(\frac{1}{x}\Big) & x\neq 0 \\ 0 & x = 0\\ \end{array} \right. $$
I think the Weierstrass' function works, but I am looking for something simpler (i.e. closed-form expressions). All elementary functions I know are only non-differentiable at "isolated" points, but not on an interval, so I cannot think of an appropriate example. Thank you in advance!
For as far as I know, all elementary functions are differentiable almost everywhere in their domain. Even adding the possibility of integrating and adding interval piecewise definitions will still result in functions differentiable almost everywhere in their domain. I expect the only way to define a continuous function not differentiable in a set of positive measure is by taking limits (or infinite sums), but then you better just refer to the Weierstrass function.