Abstract: Consider a real even function f. Even though the Euclidean norm |.| is not smooth at 0, the composite function f(|.|) is differentiable if and only if f is. The same is true for twice or three times differentiability, and there are many analogous results (for example, for the symmetric functions of matrix eigenvalues prevalent in contemporary optimization). I will outline an approach to these results via some multivariate approximation theory used in the study of Sobolev spaces.