Can it be proved that the integrand is integrable?

Question

Let ‎$x(t) \in \mathbb{R}$‎ be a continuous and derivable function. If we prove that the function $$f(\tau)=\frac{(x(t)-x(\tau))^2}{(t-\tau)^{\alpha+1}} , \alpha>0$$ is integrable, it can be shown that the following relation is established $$ \begin{equation} ‎ \int_{0}^{t} \frac{(x(t)-x(\tau))^2}{(t-\tau)^{\alpha+1}} \,d\tau \geq 0‎. ‎\end{equation}‎ $$ Can this result be obtained? In other words, it can be shown that $f$ is integrable?

Answer 1

The integral is improper at $\tau=t$. Since $x$ is differentiable at $t$, we have $$ \frac{(x(t)-x(\tau))^2}{(t-\tau)^{\alpha+1}}\sim\frac{(x'(t))^2\,(t-\tau)^2}{(t-\tau)^{\alpha+1}}\sim\frac{(x'(t))^2}{(t-\tau)^{\alpha-1}}. $$ If $x'(t)\ne0$, this is integrable at $t$ if and only if $\alpha<2$. You can get a larger range of integrability at those $t$'s with $x'(t)=0$.