Question

Let $\left[x\right]$ be greatest integer function and
$f\left(x\right)=\frac{\sin \frac{1}{4}\pi \left[x\right]}{\left[x\right]}$ Then, which of the following options is correct regarding $f\left(x\right)$:

• A. Not continuous at any point
• B. Continuous at $\frac{3}{2}$
• C. Discontinuous at $2$
• D. Differentiable at $\frac{4}{3}$

Answer 1

Answer: (B), (C), (D)

Differentiable at $\frac{4}{3}$



$f\left(x\right)=\{\begin{array}{cc}\frac{\sin \frac{\pi }{4}}{1}=\frac{1}{\sqrt{2}},& 1\le x<2\\ \frac{\sin \frac{\pi }{2}}{2}=\frac{1}{2},& 2\le x<3\end{array}$
Hence $f\left(x\right)$ is continuous at $\frac{3}{2}$, differentiable at $\frac{4}{3}$ but discontinuous at $2$.