Question

Determine the continuity of the function $f\left(x\right)=\left[x^2\right]-{\left[x\right]}^2,\forall x\in R$ at the end point of the interval $[-1,0]$. Where $[.]$ denotes the greatest integer function.

Answer 1

Let $x=n\in Z$

Left hand limit : Let h is very approx. 0

${lim}_{h\to 0^{+}}\left[\right(n+h{)}^2]-[n+h{]}^2$

${lim}_{h\to 0^{+}}{n}^2-n^2=0$

$\therefore f\left(x\right)$ continuous at$x=0$