Number of Jumps Suppose $f$ is increasing and there exist numbers $A$ and $B$ such that for all $x$, $A \leq f(x) \leq B$. Show that for each $\epsilon >0$ the number of jumps of size exceeding $\epsilon$ is at most $(B-A)/\epsilon$. Let $\epsilon >0$. So the jump size is $f(x+)-f(x-)$. So it seems that we divide the maximum possible height $B-A$ by $\epsilon$ to get the maximum number of jumps exceeding $\epsilon$ (e.g. dividing a large number by small number). Is this correct?

Since $f$ is increasing, you can't "undo" a jump upwards. Therefore $n$ jumps of size at least $\epsilon$ contribute an increase of at least $n\epsilon$, and this has to be less than the total increase $B-A$.