Problem 571 (difficulty: 5/10)

Prove that if \(\displaystyle f:[a,b]\to\R\) is convex, then \(\displaystyle \lim\limits_{a+0}f\) and \(\displaystyle \lim\limits_{b-0}f\)exists and finite, moreover \(\displaystyle \lim\limits_{a+0}f\le f(a)\) and \(\displaystyle \lim\limits_{b-0}f\le f(b)\).

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