Question

Show that the function defined by g(x)=x-[x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to x.

Answer 1

The given function is g(x) =x-[x]
It is evident that g is defined at all integral points.
Let n be an integer. 

Then,

It is observed that the left and right hand limits of f at x = n do not coincide. 

Therefore, f is not continuous at x = n 

Hence, g is discontinuous at all integral points.