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

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.

• @ViktorGlombik It's amusing to note that the very second sentence of that page reads "Not to be confused with the Intermediate value theorem"... – David C. Ullrich Nov 24 '18 at 16:09

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.
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.