If $f^2$ is differentiable, how pathological can $f$ be? Apologies for what's probably a dumb question from the perspective of someone who paid better attention in real analysis class.
Let $I \subseteq \mathbb{R}$ be an interval and $f : I \to \mathbb{R}$ be a continuous function such that $f^2$ is differentiable.  It follows by elementary calculus that $f$ is differentiable wherever it is nonzero.  However, considering for instance $f(x) = |x|$ shows that $f$ is not necessarily differentiable at its zeroes.  
Can the situation with $f$ be any worse than a countable set of isolated singularities looking like the one that $f(x) = |x|$ has at the origin?

I'd try something along these lines:
if we consider $K$ - Cantor set on $[0,1]$ and $f(x)=\inf_{y\in K} |x-y|$. This function is continuous and has a continuum of zeros: $f^{-1}(0)=K$. The only question is whether $f^2$ is everywhere differentiable. The intuition suggests that up to some tinkering with the definition of $f$ that it is.
Edit. reflected the result from my comment below.
