1. ## Proof concerning continuity

Let f:[a,b] --> R be a continuous function such that [a,b] is a subset of [f(a), f(b)]. Prove that there exists x* in [a,b] such that f(x*)=x*

2. Originally Posted by paupsers
Let f:[a,b] --> R be a continuous function such that [a,b] is a subset of [f(a), f(b)]. Prove that there exists x* in [a,b] such that f(x*)=x*
Let $g(x)=f(x)-x$ and use the intermediate value theorem on $g$.

3. Hm, I'm not following you... how will that help me prove f(x*)=x* for some x*?

I understand the "geometry" of the problem, ie, the function is "taller" than it is "wide" so the line y=x must pass through the function. Just having a difficult time proving that.

4. consider the function $g(x)=f(x)-x$, $x \in [a,b]$
because $[a,b]$ is a subset of $[f(a), f(b)]$, then $a \geq f(a)$ and $f(b) \geq b$, so $g(a) = f(a) - a \leq 0$ and $g(b) = f(b) - b \geq 0$.
From the intermediate value theorem, there is $c \in [a,b]$ such that $g(c) = 0$, that is $g(c) = f(c) - c = 0$
hence $f(c) = c$

5. Ah, thanks! That makes perfect sense!