Question

At which point the function $f\left(x\right)=\frac{x^2}{\left[x\right]}$, where $[\cdot ]$ is greatest integer function, is discontinuous?

• A. Only positive integers
• B. All positive and negative integers and $(0,1)$
• C. All rational numbers
• D. None of the above

Answer 1

The correct option is B All positive and negative integers and $(0,1)$

Clearly, if $0\le x<1$, then $f\left(x\right)$ does not exist as $\left[x\right]=0$.
Also, $\underset{x\to a}{lim}f\left(x\right)$ does not exist for any integer $a$.
Thus, $f$ is discontinuous at all integers and also in $(0,1)$.