How to prove the demicountinuity of nonlinear operators?

Question

Define a nonlinear operator $\mathbf{J}(\mathbf{x}):~\mathbb{R}^3 \rightarrow \mathbb{R}^3$ as $$ \mathbf{J}(\mathbf{x}):= |\mathbf{x}|^{-\alpha}\mathbf{x},~0<\alpha<1. $$ How to prove that $\mathbf{J}(\mathbf{x})$ is demicontinuous? Demicontinuity of an operator $\mathbf{J}$ means that $\mathbf{x}_n \rightharpoonup \mathbf{x}$ (weak convergence) implies that $\mathbf{J}(\mathbf{x}_n) \rightharpoonup \mathbf{J}(\mathbf{x})$ (weak convergence).