Question

Prove that the function $f\left(x\right)=\left[x\right]$ is not continuous at $x=0$. Where $\left[x\right]$ is the greatest integer function.

Answer 1

We've the function $f\left(x\right)=\left[x\right]$ in $[-1,1]$ is defined as

$\left[x\right]=\{\begin{array}{ccc}-1& :& -1\le x<0\\ 0& :& 0\le x<1\end{array}$.

Then,

$\underset{x\to 0+}{lim}f\left(x\right)=0$ but $\underset{x\to 0-}{lim}f\left(x\right)=-1$.

So the limit of the function $f\left(x\right)$ doesn't exist at $x=0$.

Hence the function $f\left(x\right)$ is not continuous at $x=0$.