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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!

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