Question

The function defined by $f\left(x\right)=(-1{)}^{\left[x^3\right]}$ , where [ ] denotes greatest integer function satisfies

• A. Continuous at all $x\in Z$
• B. $f(\frac{3}{2}=1)$
• C. Discontinuous for $x=n^{\frac{1}{3}},n\in Z$
• D. $f^{\prime }\left(x\right)=0for-1<x<1$

Answer 1

The correct option is C Discontinuous for $x=n^{\frac{1}{3}},n\in Z$

$f\left(x\right)=(-1{)}^{\left[x^3\right]}$

Clearly $f\left(x\right)$ is discontinuous at $x=n^{1/3},n\in Z\text{as}\left[x\right]$ is discontinuous at integral points.