$f(t)=\text{det}(A+tB)$ is continuous for symmetric positive semi-definite matrices $A$ and $B$?

Question

We have any symmetric positive semi-definite matrices $A$ and $B$ where $\text{det}B=0$.

A function $f(t)=\text{det}(A+tB)$ is continuous on $t\in[0,1]$?

$f(t_{1})-f(t_{2})=\text{det}(A+t_{1}B)-\text{det}(A+t_{2}B)$. From here, I can't go farther. Thanks in advance.

Answer 1

Expanding out $f(t)=\det(A+tB)$ using the formula for the determinant shows that $f(t)$ is a polynomial in $t$. All polynomials are continuous on all of $\Bbb R$. This is true for all matrices $A$ and $B$.