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.