TITLE: Infinitely many zeros of a nonconstant continuous function?
QUESTION [6 upvotes]: Let $f:[0,1]\to\mathbb{R}$ be a nonconstant continuous function. Is $S=\{x: f(x)=0\}$ finite? 
I have thought of a function with countably many $0$'s like lots of triangular bumps at each point $\{1/n\}$, I mean lots of $W/M$ shapes on $[0,1]$. Is it okay?

REPLY [5 votes]: $f(x)=\max\{0,2x-1\}$ is a nonconstant continuous function with uncountably many zeros.