Exercise 6 after $\S$ 91 from Paul R. Halmos's Finite-Dimensional Vector Spaces (second edition) invites to prove or disprove the following assertion.
Prove that $det$ and $tr$ are continuous.
I interpret this assertion as follows: "Prove that $det$ (determinant) and $tr$ (trace) are scalar-valued continuous mappings of the set $\mathcal L(\mathcal V)$ of all linear operators on any $n$-dimensional inner product space $\mathcal V$".
My understanding of the definitions of $det$ and $tr$ (from $\S$53. Determinants and $\S$ 55. Multiplicity respectively of the book) is as follows.
If $A$ is any linear operator on $\mathcal V$, if $\{x_1, \cdots, x_n\}$ is any basis in $\mathcal V$, and if $w$ is any non-zero alternating $n$-linear form on $\mathcal V$, then $$det(A) = \frac{w(Ax_1, \cdots, Ax_n)}{w(x_1, \cdots, x_n)}.$$
If $\mathcal V$ is defined over a complex field (algebraically closed), and if $A$ is any operator on $\mathcal V$, then $$tr (A) = \sum_{j=1}^p m_j \lambda_j,$$ where $\lambda_1, \cdots, \lambda_p$ are all the Eigenvalues of $A$, and $m_j$ is the algebraic multiplicity of $\lambda_j$ for $j = 1, \cdots, p$.
In my attempt thus far, I am able to show, hopefully correctly, that $det$ is continuous. I am clueless on how to show $tr$ is continuous however, and would appreciate a pointer.
My proof for why det is continuous: Referring to the expression for $det$, we see that the denominator, i.e., $w(x_1, \cdots, x_2)$, is a non-zero scalar (refer Theorem 3 from $\S 30$. Alternating forms). However, the numerator is a continuous scalar-valued (multilinear) mapping of the set $\mathcal L(\mathcal V)$ to which $A$ belongs. It is therefore clear that the quotient $\frac{w(Ax_1, \cdots, Ax_n)}{w(x_1, \cdots, x_n)}$ is a continuous mapping of $\mathcal L(\mathcal V)$.
My struggle with $tr$: If $\mathcal V$ is defined over the complex field, then it is clear that $tr(A)$ is the sum of all Eigenvalues (repeating if so) of $A$, whereas $det(A)$ is the product of of all Eigenvalues (repeating if so) of $A$. And, we know, hopefully correctly, that $det$ is continuous. But I haven't been able to progress further.