$\text{ The function }\mathrm{f}\left(\mathrm{x}\right)=[\mathrm{x}{]}^2-\left[{\mathrm{x}}^2\right]\text{ (where }$$\left[y\right] is the greatest integer less than$$\text{ or equal to y), is discontinuous at }$

$\begin{array}{l}f\left(x\right)={\left[x\right]}^2-\left[x^2\right]\\ \text{ Check continuity at }x=0\\ \underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }{\left[x\right]}^2-\left[x^2\right]=0\\ \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{-}}{\lim }{\left[x\right]}^2-\left[x^2\right]\\ ={\left(-1\right)}^2-0=1\end{array}$

$\begin{array}{l}\text{ Thus, discontinuous at x}=0\\ \text{ Check continuity at }x=1\\ \underset{x\to 1^{+}}{\lim }f\left(x\right)=1-1=0\\ \underset{x\to 1^{-}}{\lim }f\left(x\right)=0-0=0\\ \text{ Also }f\left(1\right)=0\\ \text{ Hence continuous at x}=1\text{ . }\\ \end{array}$