= Solution
\b[True.] Apply the preceding <Baire category theorem> argument simultaneously to $\alpha=1/j$, $j\geq1$. 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 $1/j\leq\alpha$. For $0<|h|\leq1$,
$$
|h|^{-\alpha}|f(x+h)-f(x)|\geq |h|^{-1/j}|f(x+h)-f(x)|.
$$
The right-hand limsup is infinite at every $x$, proving the assertion for every positive exponent.
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