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Let $I \subset \mathbb{R}$ be an open interval and $f: I\to \mathbb{R} $ is convex. (That is, whenever $a, b \in I$ and $0\leq t\leq 1$ then $$f((1-t)a + tb) \leq (1-t)f(a) +tf(b).$$ How can I prove that $f$ is differentiable almost everywhere? The proof of Alexandrov theorem or the contents in Rockafellar's "Convex analysis", Theorem 25.5 mentioned in Convex function almost surely differentiable. are too difficult for me, who is studying elementary real analysis. Are there any other proof of this? (I hope there is because it is the simplest case in which $ n =1$.)

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This is only a broad outline and not a detailed answer.

$f$ has right-hand and left-hand derivatives at every point. These are both monotone functions and hence they are continuous except at countable number of points. At any point where one the derivatives is continuous the function is actually differentiable. Hence $f$ is differentiable at all but countably many points, in particular almost everywhere.

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