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.

Explanation

Graph of the function is given in figure. From the graph, it seems like that f(x) is dicontinuous at every integral point of x.
Case I :
Let c be a real number which , is not equal to any 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, ` underset (x to c) lim f(x) = underset(x to c) lim [x] = [c] ` , Also f (c) = [c] and hence the function is continuous at all real number not equal to integers.
Case II : Let c be an integer. Then we can find a sufficiently small real number h gt 0 i.e. 0 lt h lt 1.
such that [c -h] = c-1 whereas =[c + h] =c
Thus, ` underset( x to c^(-)) lim f(x) = c -1 and underset(x to c^(+)) f (x) =c `
Since these cannot be equal to each other for any c, the function is discontinuous at every integral point.