Question

$f\left(x\right)=[x{]}^3-[x^3]$, (where [.] is greatest integer function) is discontinuous at all

• A. integers $n$
• B. integers $n$ except $n=0\text{and}1$
• C. integers $n$ except $n=0\text{and}1$, since $f\left(n^{-}\right)\ne f\left(n\right)$
• D. integers $n$ except $n=0\text{and}1$, since $f\left(n^{+}\right)\ne f\left(n\right)$

Answer 1

Answer: (B), (C)

integers $n$ except $n=0\text{and}1$, since $f\left(n^{-}\right)\ne f\left(n\right)$

Let $n\in$ z
$LHL=\underset{x\to n^{-}}{lim}[x{]}^3-[x^3]=(n-1{)}^3-(n^3-1)=-3n(n-1)$
$RHL=\underset{x\to n^{+}}{lim}[x{]}^3-[x^3]=n^3-n^3=0$
$f\left(x\right)$ is continuous at $n$
if $f\left(n\right)=-3n(n-1)=0$
therefore, $f\left(x\right)$ is continuous at
$n=0$ and $1$ and discontinuous at $Z-\{0,1\}$