Question

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

• A. $1$
• B. $2$
• C. non - periodic
• D. periodic but having no fundamental period

Answer 1

Answer: (D)

periodic but having no fundamental period

Consider the number line.
We know that there will be infinite points between two numbers whether the numbers are irrational or rational.
Hence there would be an irrational number in the immediate neighborhoods of a rational number and there would be rational numbers in the immediate neighborhood of an irrational number.
Hence the function will take the value 1 and 0 periodically. However, since there are infinite points on the number line and still infinite points between any two number, the period of the function cannot be determined.