Question

If $f\left(x\right)=|x-1|-\left[x\right]$, where $\left[x\right]$= the greatest integer less than or equal to $x$, then

• A. $f(1+0)=1,f(1-0)=0$
• B. $f(1+0)=-1,0=f(1-0)$
• C. ${lim}_{x\to 1}f\left(x\right)$ exist
• D. ${lim}_{x\to 1}f\left(x\right)$ does not exist

Answer 1

Answer: (B), (D)

The correct options are
B $f(1+0)=-1,0=f(1-0)$
D ${lim}_{x\to 1}f\left(x\right)$ does not exist

$f\left(x\right)=|x-1|-\left[x\right]\Rightarrow f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}1-x& 0\le x<1\end{array}\\ \begin{array}{cc}x-2& 1\le x<2\end{array}\end{array}$
Hence $f(1+0)=-1$ and $f(1-0)=0$