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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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Draw a picture and see how the secants approach the slope(s) from the left and right. – copper.hat May 25 '12 at 6:37
See e.g. van Rooij, Schikhof: A Second Course on Real Functions, Theorem 2.2, p.15 and Lemma 2.3 on the following page. – Martin Sleziak May 25 '12 at 6:37
up vote 5 down vote accepted

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 $t < 0$.

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