I'm trying to solve this question.
Let $0\leq \lambda \lt 1$ and $f:X\subset\mathbb R\to \mathbb R$ a $\lambda$-contraction, i.e., $|f(x)-f(y)|\leq \lambda |x-y|$ for any $x,y \in X$. Prove that if $X$ contains a closed interval $[a-r,a+r]$ and $|f(a)-a|\leq (1-\lambda)r$, then $f$ has a fixed point in $[a-r,a+r]$.
In order to begin to solve this question, I'm trying to prove that this function is derivable, I need help, particularly in this part.
thanks a lot