I am searching for a theorem similar to the mean value theorem for non-differentiable functions. To be exact I want to show or reference a statement similar to the following: Let $$f:\mathbb{R} \rightarrow \mathbb{R}$$ be continous and differentiable almost everywhere (differentiability everywhere except for a set of measure 0) with bounded derivative. Then it holds true that $$||f(x_1)-f(x_2)||\leq sup_{x_d \in \mathbb{R}}|f'(x_d)| \ ||x_1-x_2||,$$ where $x_d$ is a differentiable point. With other words, the optimal Lipschitz constant is given by $L = sup_{x_d \in \mathbb{R}} |f'(x_d)|$ and non-differentiable points do not need to be considered.
Of course, the inequality that $L\geq sup_{x_d \in \mathbb{R}} |f'(x_d)|$ is trivial by choosing $x_1 \rightarrow x_2$, but I have no idea how to show equality. For differentiable functions this could be done with the mean value theorem.
Please do not provide a complete proof without reference, as i need to refer to a source in my work. If there is no source, I appreciate any tipps what I could look up to be able to proof it myself. Thank you very much in advance!!
If $f$ is Lipschitz on $\mathbf R$ with Lipschitz constant $L$ then $f$ is differentiable almost everywhere with $|f'(x)| \le L$ whenever $f'(x)$ exists, and $f$ is absolutely continuous on each closed subinterval $[a,b]$. The fundamental theorem of calculus holds for AC functions. In particular $$f(b) = f(a) + \int_a^b f'(t) \, dt$$ whenever $a < b$. Consequently $$|f(b) - f(a)| \le \int_a^b |f'(t)| \, dt$$ which leads immediately to the inequality you need. This result is available in just about any textbook on real analysis.