linear algebra – How well-behaved are the operations of mapping a matrix to the $S$ or $N$ part of the $S+N$ decomposition?

Consider the maps $S,N colon mathrm{Mat}_{n}(mathbb{C}) to mathrm{Mat}_{n}(mathbb{C}) $ taking a matrix to the $S$ or $N$ part of the $S+N$ decomposition. How well-behaved are these maps? Are they e.g. continuous/Lipschitz/differentiable/smooth/holomorphic, and if not (which I suppose is the case), do they satisfy some weaker versions of these properties? What about all the other properties one could consider?