Given a function $f(x):\mathbb R\to\mathbb R$, which is continuous, bijective, and nondecreasing on $\mathbb R$. Also, there exists a constant $L>0$ such that 0<|f'(x)|\leq L for all $x\in\mathbb R$.
I want to show that
$ f((x, x+a))\subset (f(x), f(x)+La)$ for all $x\in\mathbb R$, and $a>0$, where $(x,x+a)$ is an open interval in $\mathbb R$.
Any help! Thanks.
Edit: I know from the above conditions that $f$ will be Lipschitz function with Lipschitz constant $L$, i.e., $|f(x)-f(y)|\leq L|x-y|$ and if we consider $y=x+a$, then we get $|f(x)-f(x+a)|\leq La$. But How to use this!