Hard math problem in analysis?

The answer is yes. More generally, around 1940, Zahorski completely characterized the subsets R (= the set of real numbers) that are of the form

{x in R: f is not differentiable at x}

for some continuous function f.

They are precisely the sets of the form E union F, where E is a G_delta set (a countable intersection of open sets) and F is a G_{delta sigma} set of measure zero. Aa G_{delta sigma} set is a countable union of G_delta sets. A subset S of R has measure zero if for all e > 0 there is a a countable sequence of intervals I_1, I_2, ..., with the property that S is contained in the union of the I_n, and the sum of the lengths of the I_n is less than e.

To see how Zahorski's theorem applies here let Q denote the set of rationals and R \ Q the set of irrationals. Since the rationals are countable, you can enumerate them in some order as r_1, r_2, r_3, ... , and then

R \ Q = the intersection, over all n, of the open set ((-oo, r_n) union (r_n, oo)).

This shows that R \ Q is a G_delta set, and hence by Zahorski's theorem, there is a continuous function on R whose set of nondifferentiability is precisely R \ Q. I have linked to Zahorski's paper below if you want a proof of the general result.

For an explicit example of such an f, you will have to make a mess. Examples are usually constructed using infinite series of differentiable functions that get increasingly "steep" away from the rational numbers. You can find examples in books with titles like "counterexamples in analysis"; I think a lot of older real analysis books might also include such examples. I have linked to a post on math.stackexchange.com where somebody suggests a function that *might* have this property. He does not verify the details (and I haven't checked them either). But if that function does not, something similar does. Since the construction of explicit examples involves making a lot of arbitrary choices (and ugly series expressions), many people do not find the construction of actual examples as interesting as the general proof that such examples exist.