$\text{ The function }f\left(x\right)=[x{]}^2-\left[x^2\right]\text{ (where }[y]\text{ is the tangent integer }\le y\text{ ) is discontinuous at }$

$\text{ if }0<x<1,\text{ then }0<x^2<1$

$\Rightarrow \left[x\right]=0\text{ and }\Rightarrow \left[x^2\right]=0\therefore f\left(x\right)=0\text{ for }0<x<1$

$\text{ if }1\le x<\sqrt{2},\text{ then }1\le x^2<2$

$\Rightarrow \left[x\right]=1,\left[x^2\right]=1\therefore f\left(x\right)=0\text{ for }1\le x<\sqrt{2}$

$\therefore f\text{ is continuous at }x=1$