Show that a continuous function has a fixed point Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.
I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ?

You could also nuke the mosquito: in particular, this is a special case of Kakutani's fixed point theorem.
This works because:


*

*By the closed graph theorem, the compactness of the codomain of $f : [a,b] \rightarrow [a,b]$ is sufficient to deduce the closedness of its graph.

*Given $x \in [a,b]$, the element $f(x)$ can be identified with the corresponding singleton set $\{f(x)\}$, and every singleton set is non-empty and convex.

