Let $f:\Bbb{R^2}\rightarrow\Bbb{R}$ and two real parameters $a,b$ such that $$f (\mathbf{x,y})= \begin{cases}a(x^2+y^2),&(x,y)\in B_{d_{2}}(0;2)\\ \frac{b}{\sqrt{x^2+y^2}},&(x,y)\in \Bbb{R^2}- B_{d_{2}}(0;2)\end{cases}$$ $B_{d_{2}}(0;2) $ represents the ball centered at $(0,0)$ with radius 2, with respect to the Euclidean metric $d_2$ on $\Bbb{R^2}$. We have to check the uniform continuity of $f$ on $B_{d_{2}}(0;3)$.
My attempt: I start by observing the fact that $B_{d_{2}}(0;3)$ is a compact set (Heine-Borel theorem), hence proving just continuity is enough. Any ideas on how to prove continuity?
EDIT Using polar coordinates $$f(r)=\begin{cases}ar^2,&r\in(0,2)\\ \frac{1}{r}b,&r\in\Bbb{R^+}-(0,2) \end{cases} $$ I need to check continuity in $ 0,2$ respectively. Correct? But it's not continuous.
EDIT2 After some thought we need to check $$\lim_{r\to 2 }f(r)$$ I think it all comes down to $4a=\frac{1}{2}b$. Hence f is continuous iff $b=8a$ Is it now correct?