uncontinuous sequence function convergent uniforme

Question

i wanted to know if there is some example of uncontinuous function converging uniformly to a continuous function.

Does a function converge uniformly to a bounded function must be bounded too ?

Answer 1

Consider $f_n:x\mapsto \frac 1n 1_{x\in \mathbb Q}$. Each $f_n$ is nowhere continuous but the sequence converges uniformly to the zero function (which is continuous).

With $\epsilon=1$ in the definition of uniform convergence, there is some $N$ such that $n\geq N \implies \forall x, |f_n(x)-f(x)|\leq 1$, thus $\forall x, |f_n(x)|\leq 1+|f(x)|$.
If $f$ is bounded, so are the $f_n$ for $n\geq N$.