$\alpha$ is monotonic and continuous on [a,b].
Let $\varepsilon$>0 be given then for any positive integer n,
we can choose a partition such that $\Delta\alpha_i=\frac{\alpha(b)-\alpha(a)}{n}$
since $\alpha$ is continuous. (by the theorem)
Theorem: Let f be a continuous real function on the interval [a,b].
If f(a) < f(b) and if c is a number such that f(a) < c < f(b) then there exists a point x belongs to (a,b) such that f(x)=c
I understand $\Delta\alpha_i$ can be chosen like that, but why is this justified by that theorem? I cannot find any relationship between them.