Semicontinuity of the inertia of symmetric matrices

Eigenvalues are Lipschitz. For symmetric $S,S'$ Weyl's inequality gives $|\lambda_k(S)-\lambda_k(S')|\le\|S-S'\|$ for each $k$ (min-max formula for the Rayleigh quotient).

$n_+$ is lower semicontinuous. Let $m=n_+(S)$, so $\lambda_n(S),\dots,\lambda_{n-m+1}(S)>0$. Let $\delta=\lambda_{n-m+1}(S)>0$ be the smallest of them. If $\|S'-S\|<\delta$ then
$$\lambda_{n-m+1}(S')>\lambda_{n-m+1}(S)-\delta=0,$$
and so at least $m$ eigenvalues of $S'$ are positive: $n_+(S')\ge n_+(S)$ in a neighbourhood.

$n_0$ is upper semicontinuous. The nullity is $n-\operatorname{rank}S$, and the rank is lower semicontinuous (a nonzero minor stays nonzero nearby). Hence $n_0(S')=n-\operatorname{rank}S'\le n-\operatorname{rank}S=n_0(S)$ near $S$.

Example. $S=0$ has $n_+=0$, $n_0=n$. Nearby $S'=\varepsilon I$ has $n_+=n$ and $n_0=0$: the invariants can only jump up for $n_+$ and down for $n_0$ in a neighbourhood.

Remark. By symmetry $n_-$ is also lower semicontinuous, and $n_++n_-+n_0=n$. Near a given $S$ the signature can only change by turning zero eigenvalues into positive or negative ones; $n_+$ and $n_-$ never decrease locally. In particular, $\{n_+=n\}$ (positive definite matrices) is open.