The answer is given with an idea about looking at the "jumps" a "jump" constituting ($x\_0-$point of discontinuity) $$s=\lim\_{x\to x\_0+}f(x)-\lim\_{x\to x\_0-}f(x)$$

Since $[0,1]$ is in question the largest possible "jump" is $f(1)-f(0).$ because the function is rising on $[0,1]$.

The following is unclear(given):

If we have a "jump" of length $\alpha$, then the number of points this can occur at will be **at most** is $$int \frac{f(1)-f(0)}{\alpha}$$ int- integerpart function. Can anyone explain why this is?