A problem about understanding the underlying conceptual issues of calculus.

I have a problem in understanding the underlying conceptual issues of calculus. I will use an example to illustrate my problem.

One definition of the the Mean Value Theorem is "If f is a continuous function on [a, b] (closed set) which is differentiable in (a, b) (open set), then there exists a point x belongs to (a, b) such that f(b) - f(a) = (b-a) f'(x)".

The expression that "if f is a continuous function on [a, b] which is differentiable in (a, b)" is also used in other mean value theorems as well as in other mathematical concepts.

Can someone explain the underlying rationality of this expression? Why do we need this expression so that various mean value theorems as well as other mathematical concepts can hold?

Thank you very much in advance XD