Does there exist a real function continuous everywhere and differentiable everywhere except in an uncountable subset of the reals of measure zero? I want to know if, given any function $f\colon\mathbb{R}\rightarrow\mathbb{R}$ that is continuous everywhere and differentiable everywhere except a on subset, say $S$, of $\mathbb{R}$ of measure zero, is it necessarily true that $S$ is countable? I know that there are real functions that are everywhere continuous but nowhere differentiable, but I don't know the answer to the above question. Any help would be appreciated.
 A: The best way to ask such questions is the most general.

Determine necessary and sufficient conditions on a set $D\subset
 [a,b]$ in order for there to exist a continuous function $f:[a,b]\to
 \mathbb R$ such that $D$ is precisely the set of points at which $f$
is differentiable.

Not surprising is that there is an exact answer to this.  The NASC on $D$ is that it be the intersection of a set  $A$ of type $\cal F_\sigma$ and a set $B$ of type $\cal F_{\sigma\delta}$ with the Lebesgue measure of $B$ equal to $b-a$.
If you ask instead for a continuous function of bounded variation then you can drop $A$ (remembering that the set $D$ for such functions must be of full measure).
In particular, pick your favorite $\cal F_\sigma$ set and there must a continuous function differentiable precisely at the points of that set. Expressed differently, pick your least favorite $\cal G_\delta$ set and there is a continuous function which fails to be differentiable on exactly that set.  The Cantor set is evidently allowed here, and (besides) that is the easiest example to give anyway.
See Chapter 14, Section 3. The set of points of differentiability of a function
in this monograph for a discussion:
https://www.amazon.com/Differentiation-Real-Functions-Crm-Monograph/dp/0821869906
More details are available in this Monthly survey:

A.M. Bruckner and J. Leonard, Derivatives, Amer. Math. Monthly 73 (1966), 24-56.

which is downloadable from here:
https://www.jstor.org/stable/2313749
or
here:
http://classicalrealanalysis.info/documents/andy1966.pdf
A: Cantor's function is an example of a function that is non-decreasing, continuous everywhere, and has derivative equal to zero almost everywhere. The set of points in which it fails to be differentiable is precisely the Cantor set, which is uncountable.
