To be precise the two hypothesis you need for the mean value theorem to hold are:

1)The function f is continuous in the **closed interval** a,b

2)The function f is differentiable in the **open interval** a,b

Here's the proof:

Let k be (f(b)-f(a))/(b-a)

Consider the function defined as:

g(x)=f(x)-kx

Then it is easy to show that:

g(a)= (bf(a)-af(b))/(b-a) = g(b)

By Rolle's Theorem the derivative of g must vanish on the open interval a,b. Thus there must exist a point c such that

g'(c)=f'(c)-k = 0

At this point c:

f'(c)=k

Thats it. Q.E.D