Question

The function $f\left(x\right)=\{\begin{array}{cc}0,& \text{x is irrational}\\ 1,& \text{x is rational}\end{array}$ is

• A. continuous at $x=1$
• B. discontinuous only at $0$
• C. discontinuous only at 0,1
• D. discontinuous everywhere

Answer 1

The correct option is D discontinuous everywhere

From the number theory, we already know that between any 2 rational numbers there exists an irrational number and vice versa.
Thus, for the function f(x) as defined above it will take both the values 0 and 1 in the neighbourhood of every point x = a.
Thus, function can never be continuous.