Let \(f \in \mathcal{C}[0,1]\). For every partition \(\Pi=\left\{t_{0}=00\) there is some rational partition \(\Pi^{\prime}=\left\{q_{0}=0

\[\sum_{j=0}^{n}\left|f\left(t_{j}ight)-f\left(q_{j}ight)ight|^{p} \leqslant \epsilon\]

Deduce from this that we may calculate \(\operatorname{VAR}_{p}(f ;[0,1])\) along rational points.Show \(\left|S_{p}^{\Pi}(f ;[0, t])-S_{p}^{\Pi^{\prime}}(f ;[0, t])ight| \leqslant c_{p, d} \epsilon\) using \((a+b)^{p} \leqslant 2^{p}\left(a^{p}+b^{p}ight), a, b \geqslant 0, p \geqslant 0\).