Constructing a sequence of function

Question

Construct a sequence of functions on [0,1] each of which is discontinuous at every point on [0,1] and which converges uniformly to a function that is continuous at every point?

Answer 1

$$ f_n(x)=\begin{cases} 0 & \text{if } x \text{ rational} \\ \frac{1}{n} & \text{if } x \text{ irrational} \end{cases} $$