I've had no issues employing the following theorem but there's just a slight bump that need's ironing out with the hypotheses and the domain.

Mean Value Theorem

If f(x) is continuous in the interval $\displaystyle a \leq x \leq b $ and if f'(x) exists at each value of x for which $\displaystyle a < x < b $, then there exists at least one value c of x between a and b such that $\displaystyle f(b) - f(a) = f'(c)(b - a) $

Why are the endpoints omitted for the interval of the derivative? The only thing I can come up with is that c isbetweena and b. So f can be continuous at a and b but not necessarily differentiable at a and b? But if f happened to be a straight line it would have to be differentiable at a and b.

Since continuity is contained within differentiability, then would it not be more concise to say

If f(x) is differentiable on [a,b], then there exists at least one value c of x between a and b such that $\displaystyle f(b) - f(a) = f'(c)(b - a) $