Uniform continuity of a specific function

Question

Is the function $f(a,A)\triangleq \frac{A^{-1}a}{\sqrt{a^{\star}Aa}}$ uniformly continuous? Where $a$ is a $d$-dimensional vector and $A$ is an invertible $d\times d$-matrix.

I tried to prove it but I can't get anything past pointwise... is it wrong?

Answer 1

No, it isn't.

Take $d=1$ and consider just the matrices with determinant greater than $0$. Then we are talking about the function$$\begin{array}{cccl}\mathbb{R}\times(0,+\infty)&\longrightarrow&\mathbb R\\(a,b)&\mapsto&\frac{a/b}{\sqrt{aba}}&=\frac{\operatorname{sgn}(a)}{b\sqrt{b}},\end{array}$$which is not uniformly continuous.