Let $X\subseteq \mathbb{R}$ be a subset. Let $\{f_n\}^{\infty}_{n=1}$ be a sequence of functions $f_n:X\rightarrow \mathbb{R}$, that converges uniformly to a function $f:X\rightarrow\mathbb{R}$. For each positive integer $n$, let $D_n\subseteq X$ denote the set of points at which $f_n$ is not continuous. Let $D\subseteq X$ denote the set of points at which $f$ is not continuous. Which one of the following statements is correct?
- If each $D_n$ is finite, then $D$ is finite
- If each $D_n$ has at most $7$ elements, then $D$ has at most $7$ elements
- If each $D_n$ is uncountable, then $D$ is uncountable
- None of the other three statements is correct
