Semicontinuity of the minimal polynomial degree and of the centraliser

Degree of $\mu_A$. The degree of $\mu_A$ is the smallest $d$ such that $A^d$ is a combination of $I,\dots,A^{d-1}$. So $\deg\mu_A\ge k$ iff $I,A,\dots,A^{k-1}$ are linearly independent, i.e. the $n^2\times k$ matrix $M_k(A)$ whose columns are the vectorised powers has rank $k$. This means that some $k\times k$ minor of $M_k(A)$ is nonzero. Each such minor is a polynomial in the entries of $A$, hence continuous, so
$$\{A:\deg\mu_A\ge k\}=\bigcup_{\text{minors}}\{m(A)\ne0\}$$
is open. A function $f$ with $\{f\ge k\}$ open for each $k$ is lower semicontinuous: $\liminf_{B\to A}f(B)\ge f(A)$.

Centraliser. The map $\operatorname{ad}_A:M_n\to M_n$, $X\mapsto AX-XA$, is linear in $X$ and its matrix (in the standard basis of $M_n$) depends linearly on $A$ (it is $I\otimes A-A^T\otimes I$ up to conventions). Rank is lower semicontinuous (same argument with minors: $\operatorname{rank}\ge r$ is open), so near $A$ we have $\operatorname{rank}\operatorname{ad}_B\ge\operatorname{rank}\operatorname{ad}_A$. By rank-nullity,
$$\dim C(B)=n^2-\operatorname{rank}\operatorname{ad}_B\le n^2-\operatorname{rank}\operatorname{ad}_A=\dim C(A).$$
So $\dim C$ is upper semicontinuous.

Example. $A=0$: $\deg\mu_A=1$ (minimal possible) and $\dim C(A)=n^2$ (maximal). Near $0$, nearby generic matrices have $\deg\mu=n$ and $\dim C=n$: degrees jump *up* and centraliser dimensions jump *down* off the special matrices.

Remark. Always $\dim C(A)\ge n$, with equality iff $A$ is *non-derogatory* ($\deg\mu_A=n$, i.e. $\mu_A=\chi_A$), which is an open dense condition.