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Let $I$ be an interval and $f:I\rightarrow\mathbb{R}$ be a convex function. If $x_o\in\text{Int{I}}$, then $f_R^'(x_0)$ and $f_L^'(x_0)$ both exist.

I'm a little stumped here -- I know I'm supposed to use the fact that $f$ is a convex function so that given an interval $I$, where $a,b,c\in I$ we have an $\alpha\in (0,1)$ s.t. $b=\alpha a + (1-\alpha)c$.

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    See e.g. van Rooij, Schikhof: A Second Cou$r$se on $R$eal Functions, Theorem $2$.$2$, [p.15](http://books.google.com/books?id=Cqk5AAAAIAAJ&pg=PA15) and Lemma $2$.3 on the following page.2012-05-25

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Hint: show that the difference quotients $\dfrac{f(x_0+t) - f(x_0)}{t}$ are nondecreasing as functions of $t$ for $t > 0$ and for $t < 0$.