Two continuous functions on closed interval and smallest element

Question

Assume $f,g\colon [0,1]\to [0,1]$ are continuous functions and $$ g(0)<f(0) \ \ \text{and}\ \ g(1)>f(1)$$Is it true that there exists smallest $t\in(0,1)$ such that $$ g(t)\geq f(t)$$ I know continuous will have max and min since it defined on closed interval. But I know If we could ensure such smallest element.

Answer 1

The set $\{t: g(t) \ge f(t)\}$ is closed and bounded so it has a minimum.

Answer 2

Let $h(x) = f(x) - g(x)$, $h$ is continuous.

$$h(0)= f(0) -g(0) \gt 0$$ $$h(1)=f(1)-g(1) \lt 0$$

By the intermediate value theorem there is some $x_0$ in $(0,1)$ such that $h(x_0)=0$. Also, since $h(x)=0$ is a level curve then its preimage is closed, since it is also bounded in $(0,1)$ then it is compact; so by the minimum value theorem there is such a least element.