(a) Let

be a continuous function. Suppose that

and

.

Prove that

.

Intuitively, this statement is obviously true. However, I can't seem to find a starting point for a formal proof. Where to start? Any hints?

(b) Give an example of a regulated function

such that

,

, and there is a point

such that

.

My problem here is that I don't really understand the difference between continuous and regulated functions. I do know the formal definition of a regulated function, and the fact that a continuous function is regulated but do not entirely grasp the concept.

Also, it seems rather counterintuitive to assert that there would be a point

such that

.