The number of points such that $f(x) = 0$ is finite where $f$ is continous

Question

Trying to solve some problem I think I will need the following claim: If we are given a continuous function $f: [a,b] \to \mathbb{R}$ and there doesn't exists an interval $[c,d] \subset [a,b]$ such that $f(x) = 0$ for all $x \in [c,d]$. Then the set of points $\{x \in [a,b] : f(x) = 0\}$ is finite. Is it correct? If so why?

Answer 1

There are a few things that could be clarified a little.

-First the domain of the function should be specified as bounded, otherwise there are a gambit of functions that satisfy the conditions i.e. sin(x), cos(x) etc...

-Second, we know that the set of zeros of a continuous function form a closed set. So if the condition is that the function is defined on a bounded set of the reals and has infinite number of zeros then its sequence of zeros must have a limit point by Bolzano Weierstrass. Since the set of zeros must be closed it must contain all its limit points. So as long as the function is continuous at all points of the sequence of zeros there is no problem. As we can see with Jakobian's solution where the above conditions are satisfied.