Symmetric decreasing rearrangement is sometimes continuous

This paper deals with the operation $\mathcal{R}$ of symmetric decreasing rearrangement which maps ${{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$ to ${{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$. We show that even though it is norm decreasing, $\mathcal{R}$ is not continuous for $n \geq 2$. The functions at which $\mathcal{R}$ is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiable function is co-area regular, and both the regular and the irregular functions are dense in ${{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$. Curiously, $\mathcal{R}$ is always continuous in fractional Sobolev spaces ${{\mathbf{W}}^{\alpha ,p}}({{\mathbf{R}}^n})$ with $0 < \alpha < 1$.