Real Analysis question - Show that the derivative is continuous.

[glacier302]

Homework Statement

Suppose that f is differentiable at every point in a closed, bounded interval [a,b]. Prove that if f' is increasing on (a,b), then f' is continuous on (a,b).

Homework Equations

If f' is increasing on (a,b) and c belongs to (a,b), then f'(c+) and f'(c-) exist, and f'(c-) <= f'(c) <= f'(c+).

IVT for Derivatives (also called Darboux's Theorem): Suppose that f is differentiable on [a,b] with f'(a) not equal to f'(b). If y0 is a real number which lies between f'(a) and f'(b), then there is an x0 belonging to (a,b) such that f'(x0) = y0.

The Attempt at a Solution

So I know that since f' is increasing, for any c in (a,b), f'(c-) <= f'(c) <= f'(c+). So for any h > 0, f'(c-h) <= f'(c) <= f'(c+h). If f'(c-h) is not equal to f'(c+h), then since f'(c) lies between f'(c-h) and f'(c+h), by the Intermediate Value Theorem for Derivatives there is an x0 belonging to (c-h, c+h) such that f'(x0) = f'(c).

My thought is that if I can show that the only x0 in (c-h,c+h) such that f'(x0) = f'(c) is c, then the limit of f'(x) as x approaches c must be f'(c), which means that f' is continuous at c. But how do I show that the only x0 in (c-h,c+h) such that f'(x0) = f'(c) is c?

Thank you in advance for any help!

 

[kof9595995]

I think you are on the right track. You probably can show that a increasing f' with a discountinuity would contradict with Darboux theorem.