Infinitely many zeros of a nonconstant continuous function?

Question

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?

Answer 1

Your suggestion will work well. If you want an explicit formula, look at $f(x)=x\sin(1/x)$ when $x\ne 0$, and $f(0)=0$.

Answer 2

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