On the existence of a particular type of real sequence of functions

Question

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ is continuous at points which form a proper dense subset of $\mathbb R$ ?

Answer 1

Let $f_1$ be the traditional function continuous at every irrational and discontinuous at every rational ($f(x)=1/q$ if $x=p/q$ in lowest terms, $f(x)=0$ if $x$ is irrational). Let $f_n=f_1/n$. Then $f_n\to0$ uniformly.

Answer 2

$$ f_n(t)=\frac {\lfloor t\rfloor}n. $$ Each $f_n $ is continuous on all intervals $(m,m+1) $ and discontinuous at $m $ for every integer $m $. The sequence converges uniformly to zero.