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] . $