A sequence of functions $f_n : [0, 1] → R$ which converges uniformly to a discontinuous function $f(x)$.

Question

Give an example or argue that such a request is impossible.

I argued that such a request is impossible because by theorem of the continuity of the uniform limit, if $f_n$ converges uniformly then limit function $f(x)$ is continuous. However I was wrong and I'm not sure why.

Answer 1

Such a request is possible. Take $$f_n(x)=f(x)\qquad\forall x\in[0,1]\forall n\in\Bbb N$$

Answer 2

Here's a more interesting example. Let $$f_n(x) = \begin{cases} 1/q, & x=p/q\text{ in lowest terms}, q\le n \\ 0, & \text{otherwise}\end{cases}.$$ Then $f_n\to f$ uniformly, where $f$ is what function?