I have to prove if $f: \mathbb{R} \to \mathbb{R}$ is differentiable in all the points. And $a,b$ are fixed points with $a<b$ such that $\vert f'(a) \vert <1$ and $\vert f'(b) \vert <1$ then exists $c \in (a,b)$ such that $f(c)=c$.
I defined $g(x)=f(x)-x$ then $g(a)=0$ and $g(b)=0$ and by the Rolle's Theorem exists $c \in (a,b)$ such that $g'(c)=0$ so $f'(c)=1$ and $\vert f'(c) \vert =1$ and I don't know how to use the other hypothesis
As OP did, define $g(x) = f(x) - x$. Then we have that $g'(a) < 0$ and $g'(b) < 0$.
Take $a'$ slightly larger than $a$ s.t. $g(a') < 0$, and take $b'$ slightly less than $b$ s.t. $g(b') > 0$ and s.t. $a' < b'$.
Then by the IVT, there is some $c \in (a', b') \subseteq (a, b)$ s.t. $g(c) = 0$.