Question

If $f\left(x\right)=\left[x\right]+[-x],$ then which of the following option is correct?
(where $[.]$ represents greatest integer function)

• A. $f\left(x\right)$ has missing point discontinuity.
• B. $f\left(x\right)$ has isolated point discontinuity.
• C. $f\left(x\right)$ has finite type discontinuity.
• D. $f\left(x\right)$ has infinite type discontinuity.

Answer 1

The correct option is B $f\left(x\right)$ has isolated point discontinuity.

$f\left(x\right)=\left[x\right]+[-x]=\{\begin{array}{cc}0& \text{if}x\in I\\ -1& \text{if}x\notin I\end{array}$

Clearly, the limit of the function exist at the particular point (say $a$) and the function is also defined at that point, but both are not equal, i.e.
$\underset{x\to a}{lim}f\left(x\right)$ exists but $\underset{x\to a}{lim}f\left(x\right)\ne f\left(a\right)$
So, $f\left(x\right)$ has isolated point of discontinuity at all integral points.