Let $f$ be increasing and $x_1 < \dotsb < x_n$ in $[a,b]$. Show that $\sum_{i=1}^n o(f,x_i) < f(b)-f(a)$, where $o$ is the oscillation.
Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.