# Math Help - differentiable function and fixed point

1. ## differentiable function and fixed point

HI! I've got some problem with this:

Prove that if f: R->R is differentiable and |f '(x)|<a<1 then f(x) has a fixed point.
Give and example of a function that |f '(x)|<1 doesn't guarantee the existance of fixed point.

Thanks in advence for any kind of help.

2. Are you familiar with the Contraction Mapping Theorem? If you can show that your map $f:\mathbb{R}\to \mathbb{R}$ satisfies $|f(x)-f(y)|\leq K|x-y|$ where $K<1$, then it is guaranteed to have a unique fixed point. To connect your derivative to this condition, I suggest using the Mean Value Theorem.

For a counterexample, you need to find a function that never intersects $g(x)=x$ but has a slope very close to 1 at all times.

3. I figured it out that function with that conditions is Lipschitz continuous, just using the Mean Value Theorem, with K<1. But the problem is I can't use the Contraction Mapping Theorem, you've mentioned, because I haven't got it on my calculus lessons. Are there any other possible ways? Maybe the Intermediate value theorem can help?