First slide

Functions (XII)

Question

The period of the function $f (x) = \{\begin{array}{l} 1, when x is a rational \\ 0, when x is irrational \end{array}\} is$

Moderate

A

1
B

2
C

non-periodic
D

None of these

Solution

For every rational number T, we have
$f (T + x) = \{\begin{array}{l} 1, when x is a rational \\ 0, when x is irrational \end{array}\} = f (x)$
but there is no least positive value of T for which f (T + x) = f (x) because there are infinite number of rational numbers between any two rational numbers. Therefore, f (x) is a periodic function having no fundamental period.

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