Sequence of continuous functions converging pointwise to a non-continuous function

Question

Is it possible that sequence of continuous functions is pointwise convergent to a non-continuous function?

Answer 1

Sure. Take $f_n(x) = e^{-(nx)^2}$. The limiting function $f$ is zero everywhere except at $x=0$, and $f(0)=1$ because $f_n(0)=1$ for all $n$.

In fact, each of the members of the sequence is $C^{\infty}$.

Answer 2

It's perfectly possible. The simplest example is probably $(x^n)$, which converges on $[0,1]$ to the function that is $0$ on $[0,1)$ and with value $1$ at $1$.

For the limit function $f$ to be continuous, a sufficient condition is the sequence to converge uniformly on every compact.