This seems to be just a simple limit problem and I feel like I should know it but I'm just not seeing it.
I have a continuous function f, and a fixed w
I want to show that the limit (as h goes to 0) of the absolute value of:
(1/h)*integral[ f(z)-f(w) ]dz = 0 (the integral is over a contour)
I believe the key to the problem is that f is continuous.
The Attempt at a Solution
For any a>0 there exists a b>0 such that z within b of w implies f(z) within a of f(w).
The problem is it seems to me like the 1/h term is going to infinity while the integral term is going to 0, which is indeterminate so I don't know how to get that the limit goes to 0.