Let $B_{\delta}\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B_{\delta}\to \Bbb{R}_{\geq 0}$ be real-analytic and have only one zero in $B_{\delta}$, namely, $(0,0)$. Let $\tilde{f}(r,\theta)$ be the polar coordinate representation of $f$. Let$$M(r) = \min_{\theta\in[0,2\pi)}\tilde{f}(r,\theta).$$Then, $M(r)$ is continuous for $r \leq \delta$. (It would be even cooler if $M(r)$ were also real-analytic.)
$\textbf{Ideas.}$ I'm at a loss. I'm somewhat tried thinking about the $\epsilon, \delta$ version of continuity, but it is non-trivial whenever you are dealing with polar coordinates. Any advice would be appreciate, or a counter-example.