Two continuous functions on a closed interval guarantees a fixed point?

Question

Let $f,g: [0,1] \to \mathbb{R}$ be continuous functions and assume $f(0) > g(0)$ and $f(1) < g(1)$. Prove there exists a $t \in (0,1)$ such that $f(t) = g(t)$.

So far I have tried saying as f and g are continuous then f +g are continuous but how do I incorporate the inequality?

I understand the two functions must cross and therefore their meeting point would be t. I don't know which theorems I should use to prove this.

Answer 1

Hint:Let $h(x)=f(x)-g(x)$ be a continuous function. Since $h(0) \gt 0$ and $h(1) \lt 0$, $h(x)$ must be 0 somewhere in the interval.

Answer 2

In exercises where you have continuity, equality, and want to prove that there exists an element that satisfies a certain equation, think about the intermediate value theorem.

Hint: Proving that $f(t)=g(t)$ for some $t$ means that $f(t)-g(t)=0$ for some $t$, so consider the function $h=f-g$, which is continuous.