Question

Let $f\left(x\right)=\{\begin{array}{cc}x\left[\frac{1}{x}\right]+x\left[x\right]& \text{if}x\ne 0\\ 0& \text{if}x=0\end{array}$ where $[\cdot ]$ denotes the greatest integer function, then which of the following is true?

• A. f(x) has a removable discontinuity at x=1
• B. f(x) has a non- removable discontinuity at x=2.
• C. f(x) is discontinuous at all positive integers.
• D. None of these

Answer 1

Answer: (A), (B), (C)

f(x) is discontinuous at all positive integers.

$f\left(x\right)=\{\begin{array}{cc}x\left[\frac{1}{x}\right]+x\left[x\right]& ifx\ne 0\\ 0& ifx=0\end{array}$
$RHLf\left(1^{+}\right)={lim}_{a\to 1^{+}}(x\left[\frac{1}{x}\right]+x[x\left]\right)={lim}_{x\to 1}\left(0\right)+x\left(1\right)$
$LHL={lim}_{x\to 1^{-}}(x\left[\frac{1}{x}\right]+x[x\left]\right)={lim}_{x\to 1^{-}}\left(1\right)+x\left(0\right)=1$
$f\left(1\right)=1\left[\frac{1}{1}\right]+1\cdot 1=2$.
Removable Discounting at x=1


$RHL={lim}_{x\to 2^{+}}x\left[\frac{1}{x}\right]+x\left[x\right]=2\left(0\right)+2.2=4$
$LHL={lim}_{x\to 2^{-}}\left[\frac{1}{x}\right]+x\left[x\right]=2\left(0\right)+2.1=2$
LHL $\ne$ RHL. Limit does not exist.
Non removable at x=2