Let $c(x)$ be Cantor function.
How can we prove that constant $\frac{1}{2}$ gives the best approximation in $L^1$ metric?
Let $ h(x)= \begin{cases} \frac14 \quad\text{for}\quad x\in[0,\frac13]\\ \frac12\quad\text{for}\quad x\in[\frac13,\frac23]\\ \frac34\quad\text{for}\quad x\in[\frac23,1] \end{cases} $
Will this piecewise constant function $h(x)$ give the best approximation in $L^1$? If so how to prove that?
My thoughts on this are this piecewise constant function $h(x)$ will not give the best approximation. But then how can we get the best function with three constancy intervals. Or may be somehow express $c(x)$ to get explicitly extremal problem.