I have recently read the famous Weierstrass Non differentiable function which is continuous everywhere but nowhere differentiable. 

But now my question is: 

Given a countable set $S (\subset \mathbb{R})$ , can we construct a function which is continuous everywhere but non differentiable only at points of S..? 

My Attempt:  

If $S$ finite say, $S=\{c\_1,c\_2,...,c\_n\}$ Then the following function is the desired one:$$f(x)=\sum\_{i=1}^{n}|x-c\_i|$$ 

But I cannot construct the function for a arbitrary Enumerable set $S.$ 

Can we generalize the question for an arbitrary set $S$?  

Please Help.

Thank you...!!