Which natural functions of a matrix are continuous?

Idea. A function with values in $\mathbb{Z}$ is continuous iff it is locally constant. So it suffices to find, for each integer-valued function, a convergent sequence along which the value jumps.

Answer. Only $1/\det A$ and $\operatorname{tr}A$ are continuous.

$\operatorname{tr}$ and $1/\det$. $\operatorname{tr}$ is linear. $\det$ is a polynomial, nonzero on $GL_n$, so $1/\det$ is continuous on $GL_n$.

Rank. $\frac1kI\to0$ and $\operatorname{rank}\frac1kI=n$ for all $k$, but $\operatorname{rank}0=0\neq n$. Not continuous (it is only lower semicontinuous: Problem 83).

$\dim\ker$. By rank-nullity $\dim\ker A=n-\operatorname{rank}A$, so the same sequence gives $\dim\ker(\frac1kI)=0\not\to n=\dim\ker0$.

Number of distinct real eigenvalues. Take $n=2$ and
$$A_\varepsilon=\begin{pmatrix}0&-\varepsilon\\\varepsilon&0\end{pmatrix}.$$
Its characteristic polynomial is $t^2+\varepsilon^2$, with no real root for $\varepsilon\ne0$, so the count is $0$. As $\varepsilon\to0$, $A_\varepsilon\to0$, which has the real eigenvalue $0$: count $1$. The count jumps. (For $n\ge2$ use block diagonal matrices.)

The jump can also go down. $\operatorname{diag}(1,1+\varepsilon)\to I$ has $2$ distinct real eigenvalues for $\varepsilon\neq0$ but $I$ has $1$. So the count is neither upper nor lower semicontinuous.

Remark. Counting eigenvalues *with multiplicity in $\mathbb{C}$* gives the constant $n$, and the multiset of complex eigenvalues does depend continuously on $A$ (Problem 81).