Points of discontinuity of the greatest integer function f(x) = [x], where [x] denotes integer less than or equal to x, are

all integers

$f(x)=[x]$

so for $0 ~f(0)=0 $

$0.3 ~f(0.3)=0 $

$f(1.2)=1$

behaviour of greatest 

Let O be integer

So $\lim\limits_{x→0^-}f(x) = \lim\limits_{h→0}f(0-h)=-1$

while $\lim\limits_{x→0^+}=\lim\limits_{h→0}f(0+h)=0$

points of discontinuity 

for any integer K

$\lim\limits_{x \rightarrow k^{-}} f(x)=\lim\limits_{h \rightarrow 0} f(k-h)=k-1$

$\lim\limits_{x \rightarrow k^{+}} f(x)=\lim\limits_{h \rightarrow 0} f(k+h)=k$

so points of discontinuity exist for all integers