Proving Convexity of a Function on an Open Interval

[Jonas Rist]

Hello,

how can I proof this:
given: a convex function on an open interval I,which is a subset of R.
I have to show that the function can be differentiated on the whole interval.
I already proved the following for a<b<c, a,b,c in I:
f(b)-f(a))/(b-a)<(f(c)-f(a))/(c-a)<(f(c)-f(b))/(c-b).

Thanks for help!
Jonas

 

[HallsofIvy]

Isn't f(x)= |x| convex on I= (-1,1)?

 

[Jonas Rist]

Ah, sorry,
actually one has to show that the function is differentiable from the left and from the right(but these derivatives don´t have to be equal, as your example shows clearly).
Jonas