Analysis Midterm

Well I had my undergraduate analysis midterm today. I think it is pretty safe to assume that I dominated that test. When I first saw the test I began to worry, but as I started to work on the problems thing began to settle down. The test only covered material from the chapters on differentiation and integration.
Here are the problem statements as I remember them, I might post my solutions later on.

1:
If for some interval , show that is uniformly continuous on this interval. (Note that need not be closed, or bounded)

2:
part 1: Suppose that is continuous at . Prove that is differentiable at .
part 2: Define a function by if and if . Find the value of and prove your result.

3:
Define when and 0 at the origin. What is the minimum value of that causes to be bounded. Also for this is continuous at 0? Prove your result. Finally characterize the values of such that is continuous on .

4:
Prove that if and only if for every there exist a partition such that .

5:
Suppose that is continuous on , and . Prove that,
.

6:
Let be differentiable on the closed interval such that . Prove that there exist a such that .

As you can see, nothing really difficult about the exam. I don't know why I panicked at the beginning, maybe I was anxious or something.