Question

Let [x] denote the greatest integer less than or equal to x. If f(x) = [x sinπ x], then f(x) is

• (A) continuous at x = 0
• (B) continuous in (-1, 0)
• (C) differentiable at x = 1
• (D) differentiable in (-1, 1)

Answer 1

The Correct Option is A, D

Explanation:
We have, for \(-1$\therefore [x\sin \pi x]=0$Also, $x\sin \pi x$ becomes negative and numerically less than 1 when $x$ is slightly greater than 1 , and so by definition of $[x]$ $f(x)=[x\sin \pi x]=-1,$ when \(1Thus, $f(x)$ is constant and equal to 0 in the closed interval [-1,1] and $s0f(x)$ is continuous and differentiable in the open interval (-1,1) At $x=1,f(x)$ is discontinuous. since $\underset{i\to 0}{lim}(1-h)=0$ and $\underset{t\to 0}{lim}(1+h)=-1$$\therefore f(x)$ is not differentiable at $x=1$ Hence, options (1),(2) and (4) are correct answers.