**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?