Continuous function whose $\{x\in(a,b):f(x)>0\}$ cannot be expressed as the finite union of open intervals

Question

Let $f$ be a continuous function on an open interval $X=(a,b)$. I know $\{x\in X: f(x)>0\}$ is open, and open sets are countable unions of disjoint open intervals. I am wondering if we can say that $\{x\in X: f(x)>c\}$ is a finite union of disjoint intervals.

If yes, how can we prove it? If no, what is the counterexample?

Thanks in advance!