Question

prove that the greatest integer function x is continuous at all points except at integer points

Answer 1

Given: The greatest integer function x .

To find: Prove that the greatest integer function x is continuous at all points except at integer points.

Solution:

• Now we have given greatest integer function x, f(x) = [x]. Consider x = 2.

• Checking its continuity , we get:

                       f(x) = [2] = 2               ................. (i)

• Now we have left side limit :

                       $\lim_{x \to \ 2-h} f(x) = [2 - h] = 1$  ...................(ii)

• Now 2 - h lies in between 1 & 2 and the least can be 1.

• Now we have right side limit :

                       $\lim_{x \to \ 2-h} f(x) = [2 - h] = 2$ ...................(iii)

• Now 2 + h lies between 2 & 3 and the least can be 2.

• Now considering (i), (ii), and (iii), left side limit is not equal to right side limit.  So limit of the function does not exist.

• So the function is discontinuous at x = 2 .

• Now from this example, Greatest integer function f(x) is not continuous at all integral points.

• Hence proved.

Answer:

                So we proved that the greatest integer function x is continuous at all points except at integer points.