TITLE: Suppose $y(x)$ is continuous and $y'(x)=0$ has uncountable many solutions but $y(x)$ is not constant on any interval. Is this possible?
QUESTION [5 upvotes]: Suppose $y(x)$ is continuous and $y'(x)=0$ has uncountable many solutions but $y(x)$ is not constant on any interval. Is this possible?

REPLY [3 votes]: Take a Cantor set $C\subseteq [0,1]$ and let $d(x)$ be the distance function from $x$ to $C$.  Then $(d(x))^2$ is differentiable with derivative vanishing at all points of $C$ hence uncountably many.