Question

The set of all points for which $f\left(x\right)=\frac{|x-3|}{|x-2|}+\frac{1}{1+\left[x\right]}$ is continuous, is $($where $[\ast ]$ represents the greatest integer function$)$

• A. $R-[-1,0]$
• B. $R$
• C. $R-\left\{\right(-1,0)\cup n,n\in I\}$
• D. $R-\left(\right\{2\}\cup [-1,0])$

Answer 1

The correct option is C $R-\left\{\right(-1,0)\cup n,n\in I\}$

$f\left(x\right)=\frac{|x-3|}{|x-2|}+\frac{1}{1+\left[x\right]}$
$x\ne 2$ and $1+\left[x\right]\ne 0\Rightarrow \left[x\right]\ne -1$
$\therefore x\notin [-1,0)$
And $\left[x\right]$ is disjoint at every integer
So, $f\left(x\right)$ is continuous for $x\in R-\left\{\right(-1,0)\cup n,n\in I\}$