Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$. Show that $H$ is continuous on $]0, +\infty[$.
Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that $H$ is continuous on $]0, +\infty[$.