Say we have a power series of two variables, with an associated function $f$ defined as $$ \begin{split} f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\ & a_{n,m} \geq 0 \quad \forall n, m \in\mathbb{N}, \end{split} $$ which is known to converge in each point of a compact set $\mathcal{C} \subset \mathbb{R_+}^2$. We can also assume $f$ to be bounded on $\mathcal{C}$.
Can $f$ have a point of discontinuity (or more) on $\mathcal{C}$?
Edit : As shown by Tom Goodwillie with this nice counter-example, it turns out to be possible for $f$ to be discontinuous. Thanks!
Edit : clarified question as per the inputs from Tom Goodwillie, David E Speyer and Daniele Tampieri.