Mathematics - Continuity - Differentiability Question with Solution | TestHub

Given,

$f\left(x\right)=\left|\left[x\right]\right|+\sqrt{x-\left[x\right]}$

$x-\left[x\right]\ge 0\Rightarrow x\in R$

Now using the property of greatest integer function and simplifying the function we get,

$f\left(x\right)=\left\{\begin{array}{c}2+\sqrt{x+2}, -2<x<-1\\ 1+\sqrt{x+1}, -1\le x<0\\ \sqrt{x}, 0\le x<1 \end{array}\right.$

Now plotting the diagram we get,

Now from the diagram we can say that, $f\left(x\right)$ is discontinuous at two points $x=\left\{–1, 0\right\}$.