Question

The function $f\left(x\right)=[x{]}^2-[x^2],$ where $\left[y\right]$ is the greatest integer less than or equal to $y$, is discontinuous at

• A. all integers
• B. all integers except $-1$
• C. all integers except $0$
• D. all integers except $1$

Answer 1

The correct option is D all integers except $1$

Given, $f\left(x\right)=[x{]}^2-[x^2]$
Let $x=n$ where $n\in Z$
$\underset{x\to n^{+}}{lim}f\left(x\right)=n^2-n^2=0$
$\underset{x\to n^{-}}{lim}f\left(x\right)=(n-1{)}^2-(n^2-1)=-2n+2f(n)=0$

For the function to be continuous at $x=n,$
$-2n+2=0\Rightarrow n=1$
Hence, function is discontinuous at all integers except $x=1.$