Could non-continuous sequence of functions converge uniformly to non-continuous function?

Question

I know there is a non-continuous sequence of functions $f_n(x)$ which is converge uniformly to a continuous function $f(x)$

My question if there exist a non-continuous sequence of functions $f_n(x)$ that converge uniformly to non-continuous function $f(x)$ ?

Answer 1

Yes. Take for example $f_n(x)=\lfloor x\rfloor+\frac{1}{n}$ which converge uniformly to $f(x)=\lfloor x\rfloor$. Actually you can take any not-continuous function $f$, any sequence $(a_n)_n$ such that $a_n\to 0$ and let $f_n=f+a_n$.