Discontinuous and converge uniformly

Question

constract a sequence of functions on [o,1] each of which is discontinous at every point on [0,1] and which converges uniformly to a function that is continous at every point

Answer 1

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

Answer 2

$$f_n(x) =\begin{cases} \frac1n, & x \notin \mathbb{Q} \\0, & x \in \mathbb{Q} \end{cases}.$$ This sequence of function converges uniformly to the function $f(x) =0$ for all $x$.