Suppose that $\{f_n\}$ is a sequence of continuous functions on $X$ s.t. $f_n\to f$ for all $x\in D$, where $D$ is a dense subset of $X$ and $f$ is continuous. Is it true that $f_n\to f$ for all $x\in X$.
Let $y\notin D$ and suppose that $f_n(y)\to g(y)$. Then $g(y)$ must be equal $f(y)$. How does one show that $f_n(y)$ converges?
No, it is not true. Take $X=[0,1]$, $D=[0,1)$, $f(x)=0$ and $f_n(x)=x^n$. Then$$(\forall x\in D):\lim_{n\to\infty}f_n(x)=f(x),$$but $\lim_{n\to\infty}f_n(1)=1\neq f(1)$.