Question

If $f\left(x\right)=\left[\text{sin}\pi x\right]$ in $(-\frac{1}{2},\frac{1}{2}),$ then which of the following(s) is/are correct:
where $\left[x\right]$ denotes the greatest integer less than or equal to $x.$

• A. $f\left(x\right)$ is continuous at $x=0$
• B. $f\left(x\right)$ is continuous in $(-1/2,0)$
• C. $f\left(x\right)$ is continuous in $(-1/2,1/2)$
• D. $f\left(x\right)$ is continuous in $(0,1/2)$

Answer 1

Answer: (B), (D)

$f\left(x\right)$ is continuous in $(0,1/2)$

$f\left(x\right)=\left[\text{sin}\pi x\right],x\in (-\frac{1}{2},\frac{1}{2})=\{\begin{array}{cc}-1,& x\in (-1/2,0)\\ 0,& x\in [0,1/2)\end{array}$

Graph of $\left[\text{sin}\pi x\right]$ in $(-1/2,1/2)$

Thus, $f\left(x\right)$ is continuous in $(-\frac{1}{2},0)$ and $(0,\frac{1}{2})$