Find all the points of discontinuity of the greatest integer function defined by $f(x) = [x]$,where $[x]$ denotes the greatest integer less than or equal to $x$.

(N/A) First,observe that $f$ is defined for all real numbers. The graph of the function is shown in the figure. From the graph,it appears that $f$ is discontinuous at every integral point. Below,we verify if this is true.
Case $1$: Let $c$ be a real number which is not an integer. It is evident from the graph that for all real numbers close to $c$,the value of the function is equal to $[c]$; i.e.,$\lim_{x \to c} f(x) = \lim_{x \to c} [x] = [c]$. Also,$f(c) = [c]$,and hence the function is continuous at all real numbers that are not integers.
Case $2$: Let $c$ be an integer. Then we can find a sufficiently small real number $r > 0$ such that $[c - r] = c - 1$,whereas $[c + r] = c$.
This,in terms of limits,means that:
$\lim_{x \to c^-} f(x) = c - 1$ and $\lim_{x \to c^+} f(x) = c$.
Since these limits are not equal to each other for any integer $c$,the function is discontinuous at every integral point.