Symmetric decreasing rearrangement is sometimes continuous

This paper deals with the operation (Equation presented) of symmetric decreasing rearrangement which maps W^{1, p}(Rⁿ) to W^{1, p}(Rⁿ). We show that even though it is norm decreasing, (Equation presented) is not continuous for n ≥ 2. The functions at which 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 W^{1, p}(Rⁿ). Curiously, (Equation presented) is always continuous in fractional Sobolev spaces W^{α, p}(Rⁿ) with 0 < a < 1.