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Homework Statement
Prove that if f(x) is convex on an open interval, then left and right derivatives of f exist at every point.
The Attempt at a Solution
So i have been able to show (from the definition of convex) that for any x1< x < x2 in the interval:
[tex]\frac{f(x)-f(x_1)}{x-x_1}\leq \frac{f(x_2)-f(x_1)}{x_2-x_1} \leq\frac{f(x_2)-f(x)}{x_2-x}[/tex]
in other words, the differential quotient in increasing as x1 increases towards x and decreasing as x2 approaches x.
i'm not sure how to incorporate these facts.
also, should my proof somehow involve that any convex function on an open interval in continuous?