Theorem. Let , and let be an interval containing . Let . Suppose that the function is integrable on for all with and on for all with . Let be the area accumulation function for on based at . Then is uniformly continuous on . If happens to be continuous on , then is differentiable on and for all .
Proof. Suppose , an interval containing , and let be a function. The area accumulation function is defined by . (1) We want to show that is uniformly continuous on . To do this, would we use a sequence approach? Because is not Lipschitz. (2) Suppose is continuous on . We want to show that is differentiable at some point . So we look at the following:
We want to show that . And so . The result follows.
Is the second part correct? In the first part, would the easiest way be to use a sequence approach? Or just the regular definition?