RO-NMO 2000 Problem 122

For any partition $P$ of $[0,1]$, consider the set $$\mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f'\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} .$$ Prove that there exists a partition $P_0$ of $[0,1]$ such that $$g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g'(x)\big| >4\cdot \# P.$$

Here, $\# D$ denotes the natural number $d$ such that $0=x_0<x_1<\cdots <x_d=1$ is a partition $D$ of $[0,1] .$