Question

Let $f\left(x\right)=\left[x^2\right]-{\left[x\right]}^2$, where [.] denotes the greatest integer function. Then

• A. $f\left(x\right)$ is discontinuous for all integral values of $x$
• B. $f\left(x\right)$ is discontinuous only at $x=0,1$
• C. $f\left(x\right)$ is continuous only at one integral value of $x$ which is $x=1$
• D. none of these

Answer 1

Answer: (C)

$f\left(x\right)$ is continuous only at one integral value of $x$ which is $x=1$

$f\left(x\right)=\left[x^2\right]-{\left[x\right]}^{2}atx\to 1^{-}f\left(x\right)=0atx\to 1^{+}f\left(x\right)=1-1=0socontinuousat1$
Hence the answer is C