The problem I am faced with is the following. Let $\{a_n\}$ be a sequence of points that is dense in some bounded open set $U \subset \mathbb{R}^n$, $n \geq 2$. Let $f_n : U \to \mathbb{R}$ be a function that is continuous everywhere except at the points $\{a_1, a_2, \ldots, a_n\}$. Is the pointwise limit function $f(x) := \lim_{n\to \infty} f_n(x)$ continuous at some point in $U$?
I thought about trying to apply the corollary to the Baire category theorem which states that if $f_n : U \to \mathbb{R}$ is a sequence of continuous functions then the pointwise limit is continuous on a generic subset of $U$ (i.e., the set of points where the pointwise limit is discontinuous is of first category). However, for that to work you need continuity at every point, whereas my functions are discontinuous on a (growing) finite number of points.
Edit: The actual function I am concerned with is $f(x) = \sum_{n=1}^{\infty} \frac{\log||x-a_n||}{2^{n}}$, where again $f$ is defined on an open set $U$ of $\mathbb{R}^n$ and $\{a_n\}$ is a sequence dense in $U$. Here $f_n$ are the partial sums. At points in the sequence $\{a_n\}$, $f(a_n)$ is defined to take the value $-\infty$.
Let $(a_n)$ be the sequence of points with rational coordinates, $f_n(a_i)=1$ for $1 \leq i \leq n$ and let $f_n$ be $0$ at all other points. Then the hypothesis is clearly satisfied. The limit function is not continuous at any point.