Question

If [.] denotes greatest integer function and f(x) = [x] $\left\{\frac{sin\frac{\pi }{[x+1]}+sin\pi [x+1]}{1+\left[x\right]}\right\},$ then

• A. f(x) is continuous in R
• B. f(x) is continuous but not differentiable in R
• C. f’(x) exists $\forall xϵR$
• D. f(x) is discontinuous at all integral points in R

Answer 1

The correct option is D

f(x) is discontinuous at all integral points in R

AT $x=n,f\left(n\right)=\frac{n}{n+1}sin\left(\frac{\pi }{n+1}\right)=f\left(n^{+}\right)$

$f\left(n^{-}\right)=\frac{n-1}{n}sin\frac{\pi }{n}$

$\Rightarrow f\left(x\right)$ is discontinuous at all integral points.