Question

Which of the following functions is periodic?

• A. f(x) = x – [x] where [x] denotes the largest integer less than or equal to the real number x
• B. f(x) = sin(1/x) for x ≠ 0, f(0) = 0
• C. f(x) = x cos x
• D. None of these

Answer 1

Answer: (A)

f(x) = x – [x] where [x] denotes the largest integer less than or equal to the real number x

Answer 2

f(x) = x – [x] = $\left{ \begin{matrix} ... \ \begin{matrix} x - 1 \ x - 2 \ x - 3 \end{matrix} \ ... \end{matrix}\begin{matrix} \ \begin{matrix} , \ , \ , \end{matrix} \

\end{matrix}\begin{matrix} \ \begin{matrix} 1 \leq x < 2 \ 2 \leq x < 3 \ 3 \leq x < 4 \end{matrix} \

\end{matrix} \right.\ $

Graph of function is

Clearly it is a periodic function with period 1.

∴ (1) is the correct alternative.