Which of the following is/are periodic?

PHYSICS · Multi-part · ISC · Physics · Oscillations and Waves · SHM kinematics

Problem

Which of the following is/are periodic? (a) f(x)={1,if x is rational0,if x is irrationalf(x) = \begin{cases} 1, & \text{if } x \text{ is rational} \\ 0, & \text{if } x \text{ is irrational} \end{cases} (b) f(x)={x−[x],2n≤x<2n+112,2n+1≤x<2n+2f(x) = \begin{cases} x - [x], & 2n \le x < 2n+1 \\ \frac{1}{2}, & 2n+1 \le x < 2n+2 \end{cases} , where [x][x] denotes the greatest integer function (c) f(x)=(−1)[x]f(x) = (-1)^{[x]} , where [x][x] denotes the greatest integer function (d) f(x)=ax−[ax+a]+tan⁡(πx2)f(x) = ax - [ax + a] + \tan\left(\frac{\pi x}{2}\right) , where [x][x] denotes the greatest integer function

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