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](https://mathoverflow.net/a/426980/102634), 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.



Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?
The series $f(x,y)=y+xy+x^2y+x^3y+\dots$ converges to $0$ when $y=0$, and converges to $y/(1-x)$ when $|x|<1$. This function is not continuous at $(x,y)=(1,0)$.


