Let $f\left(x\right)=\left[x\right]+[-x]$ where, [x] denotes the greatest integer not greater than $x$. Then for any integer $m$ 

For $m-1<x<m,\left[x\right]=m-1\text{ and }[-x]=-m$

 and for  $m<x<m+1,\left[x\right]=m\text{ and }[-x]=-m-1\text{. }$ Thus

$\underset{x\to m}{\lim }f\left(x\right)=m-1-m=-1\text{ and }\underset{x\to m+}{\lim }f\left(x\right)=m-m-1=-1$

Hence lim $f\left(x\right)$ exists but is not equal to $f\left(m\right)$