if $\operatorname{m}_n(\mathbb{c})$ is the set of $n \times n$ complex matrices and $a \in \operatorname{m}_n(\mathbb{c})$, then we write $\sigma(a) \subseteq \mathbb{c}$ for the set of eigenvalues of $a$. if $a$ is diagonalizable and $f \colon \sigma(a) \to \mathbb{c}$ is any function, then one can define $f(a) \in \operatorname{m}_n(\mathbb{c})$ in a reasonable way. now, let $\operatorname{m}_n(\mathbb{c})_{\operatorname{sa}}$ be the set of $n \times n$ hermitian matrices, which are unitarily diagonalizable and have real eigenvalues. if $f \colon \mathbb{r} \to \mathbb{c}$ is a continuous function, then one can fairly easily show that the map $\tilde{f} \colon \operatorname{m}_n(\mathbb{c})_{\operatorname{sa}} \to \operatorname{m}_n(\mathbb{c})$ defined by $a \mapsto f(a)$ is also continuous. in this talk, we shall discuss the less elementary fact that if $f$ is $k$-times continuously differentiable, then so is $\tilde{f}$. time permitting, we shall also discuss the much more complicated infinite-dimensional case -- where instead of matrices, one considers linear operators on a hilbert space -- which is still an active area of research.