# Every $f : [a, b] → [a, b]$ has a fixed point where $f$ is continuous. Deduce the intermediate value theorem.

Every $f : [a, b] → [a, b]$ has a fixed point and $f$ is continuous (on $[a,b]$). Deduce the intermediate value theorem.

I managed to show the other way, now I'm here.

I know that $f(c)=c$ for some $c\in [a,b]$, and I need to show that for all $x\in [f(a),f(b)]$ there is $f(y)=x$.

I know that $a<f(a),f(b)<b$ and $f(c)=c$.

Can somebody give me a hint?

• We can assume $f$ to be continuous with $f(a) f(b) < 0$ and we need to find $c$ with $f(c) = 0$. This appears tricky as we need to find a function $g$ such that $g(c) = c$ implies $f(c) = 0$. This is possible if $g(x) = f(x) + x$. The challenge is to find an interval $I \subseteq [a, b]$ such that $g(I) \subseteq I$. Commented May 8, 2016 at 16:32
• The word "every" is very important here; also, I believe the problem is not to prove the intermediate value theorem for only functions $f:[a, b] \rightarrow [a, b]$. I think it is to prove the intermediate value theorem in general Commented May 8, 2016 at 16:39
• Notice also that the $[f(a), f(b)] \subset f([a,b])$,and the latter is what you want for the intermediate value theorem
– Ant
Commented May 8, 2016 at 16:48
• Do you know that every continuous function is bounded? Commented May 8, 2016 at 16:59
• Be that as it may, here's a big hint: Show that if $f$ is an arbitrary continuous, real-valued function on $[a, b]$, and if $c$ is a real number between $f(a)$ and $f(b)$, then there exists a non-zero real number $\eta$ such that the function$$g(x) = x + \eta\bigl(f(x) - c\bigr),\quad x \in [a, b],$$maps $[a, b]$ into $[a, b]$. Now apply the fixed-point property to $g$. I've written up the details to my satisfaction (i.e., I'm sure this approach works), but hate to deprive you the pleasure of working things out yourself (which requires some creativity). :) Commented May 8, 2016 at 17:28