Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 23F / d / Solution

Use the centered Hardy-Littlewood maximal functionIf $x>0$, choosing $0<r<x$ makes the whole interval lie in the positive half-line, so $M\theta (x)=1$. If $x=0$, every centered interval has exactly half its length in the positive half-line, so $M\theta (0)=1/2$.

If $x<0$, the average is zero for $r\le -x$. For $r>-x$, it isand these values tend to $1/2$ as $r\to \infty$. HenceFor the uncentered maximal-function convention, intervals extending arbitrarily far to the right make the supremum equal to $1$ at every $x$.