Let f : ℝ → ℝ be defined as follows
\(f(x)=\left\{\begin{array}{c} 1, \text { if } x=0 \\ 0, \text { if } x ∈ \mathbb{R} \backslash \mathbb{Q} \\ \frac{1}{n}, \text { if } x=\frac{m}{n} \end{array}\right.\)
where m, n ∈ ℤ n > 0 gcd (m, n) = 1
Then
1
f is continuous everywhere except 0
2
f is continuous only at the irrational.
3
f is continuous only at the non-zero rational.
4
f is nowhere continuous.