Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 10I / i / Solution

True. Apply the preceding Baire category theorem argument simultaneously to $\alpha =1/j$, $j\ge 1$. The union of all the corresponding $E_{m,j}$ is still a countable union of closed sets with empty interior, so its complement is nonempty. For any $\alpha >0$, choose $j$ with $1/j\le \alpha$. For $0<|h|\le 1$,The right-hand limsup is infinite at every $x$, proving the assertion for every positive exponent.