Question

The function $f\left(x\right)=\left[x\right]-\left[x^2\right]$ (where $\left[x\right]$ is the largest integer $\le x$ ) is discontinuous at

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

Answer 1

The correct option is D all integers except $1$

Clearly, $f\left(x\right)=0$ for each integral value of $x$
Also if $0<x<1$, then $0<x^2<1$
$\Rightarrow \left[x\right]=0\&\left[x^2\right]=0$
Therefore, $f\left(x\right)=0$

Also if $-1<x<0$, then $0<x^2<1$
$\Rightarrow \left[x\right]=-1\&\left[x^2\right]=0$
Therefore, $f\left(x\right)=-1$

Also if $1<x<\sqrt{2}$, then $1<x^2<2$
$\Rightarrow \left[x\right]=1\&\left[x^2\right]=1$
Therefore, $f\left(x\right)=0$ from $0$ to $\sqrt{2}$
So $f\left(x\right)$ is continuous at $1$

So it is not continuous at any integer values except $1$ as $f\left(x\right)$ is not $0$ in other intervals between integers.
Hence, answer is "D"