Problem :
How to find the function $f(x) = [x]^2-[x^2]$ is discontinuous at all integers except 1. where $[\hspace{2pt}\cdot\hspace{2pt}]$ represents the greatest integer function.
My approach : let $x = n \in Z$
Left hand limit : Let h is very approx. 0
$\lim_{h \to 0^+} [n+h]^2-[(n+h)^2]$
$\lim_{ h\to 0^+} n^2-n^2 = 0$
Now how to proceed further please suggest will be of great help , thanks.
Best Answer
For h tending zero from above, we have a limit of zero as shown in your attempt.
For h tending to zero from below we get $lim_{h→0^-}[n+h]^2 = lim_{h→0^-}(n-1)^2 = n^2 -2n +1$ and $lim_{h→0^-}[(n+h)^2] = lim_{h→0^-}[n^2 +2hn +h^2] = n^2 -1$ for sufficiently small h. Thus $lim_{h→0^-}[n+h]^2-[(n+h)^2]= 2-2n.$
These two limits differ form $n \neq 1$. Thus $f(x)$ discontinuous at all integers except 1.