"Switching" of a continuous function

Question

Let $f, g: [0,1] \rightarrow \mathbb{R}$ be continuous functions and consider the continuous function $h(x) = \max\{f(x), g(x)\}$. Suppose that at $0$, $f$ is "active" and at $1$, $g$ is active. That is $h(0) = f(0) \neq g(0)$, and similarly at $x = 1$. Is it true that there must exist some point $t \in [0,1]$ such that $h$ "switches" from $f$ to $g$? If not I think this should violate continuity but I don't see exactly how.

Answer 1

Consider $u=g-f$. $u$ takes a negative value at $0$, a positive one at $1$, and is continuous, so...