= Solution
For $h\ne0$, enlarge the nonnegative integral $I(h)$ to all of $\mathbb R$, and substitute $y=x+|h|t$. Reflection if necessary gives
$$
I(h)\leq |h|^{1-2\alpha}C_\alpha,\qquad C_\alpha=\int_{\mathbb R}\bigl(|t-1|^{-\alpha}-|t|^{-\alpha}\bigr)^2\,dt.
$$
Near $0$ and $1$, its integrand is bounded by a constant times $1+|t|^{-2\alpha}$ or $1+|t-1|^{-2\alpha}$, which is integrable. For large $|t|$, the <mean value theorem> bounds the difference by a constant times $|t|^{-\alpha-1}$, whose square is integrable. Thus $C_\alpha<\infty$. The preceding estimate yields
$$
\boxed{|g(x+h)-g(x)|\leq\|f\|_2\sqrt{C_\alpha}\,|h|^{1/2-\alpha}.}
$$
In particular \b[$g$ is continuous on all of $\mathbb R$]; the proof establishes <Hölder continuity of a one-dimensional fractional kernel integral> of exponent $1/2-\alpha$ uniformly in $x$.
Back to article page
