the mean value theorem assumes a STRONGER condition on f than just continuity, namely that of differentiability. you also need continuity on the "larger" interval [x,α], or the mean value theorem may be FALSE, for example:

let f(x) = x on [0,1)

f(1) = -1.

then (f(1) - f(0))/(1-0) = -1, but even though f(x) is differentiable on (0,1), there is no c in (0,1) where f'(c) = -1.

in fact, one can prove that differentiability implies continuity, which puts a high degree of circularity in your proof.

my question is: what do you mean by "continuous" if not:

?