Let $f:[0,1]\rightarrow[a,b]$, $f(\cdot)$ is continuous and has no flat regions. Can the equation $f(x)=c$ have an infinite number of solutions? It seems pathological cases like the Weierstrass function might provide some examples of this, but I'm getting confused trying to understand if the Weierstrass function works here.
The other thing is, if there is an infinite sequence of solutions in $[0,1]$, it must have a convergent subsequence, converging to, say $x^*$. By continuity, $f(x^*)=c$. But also, by infinite solutions, for any $\epsilon$, there are infinite solutions in $(x^* - \epsilon, x^* + \epsilon)$, so $f(\cdot)$ is oscillating a lot around $x^*$ (like $sin(\frac{1}{x})$ close to 0). Would that somehow violate continuity? I don't think that's necessarily true, but unable to find an example.
