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