Question

Show that the greatest integer function f(x) =[x] is continuous at all points except at integer points

Answer 1

Dear Student,
Lets take an example to check its continuity at x = 2.
i) f(x) = [x], for all x in R
==> By the definition of greatest integer function: If x lies between two successive integers, then f(x) = least integer of them.

ii) So, at x = 2, f(x) = [2] = 2 -------- (1)

Left side limit (x ---> 2-h): f(x) = [2 - h] = 1 ----- (2)
{Since (2 - h) lies between 1 & 2; and the least being 1}

Right side limit (x --> 2+h): f(x) = [2 + h] = 2 ---(3)
{Since (2+h) lies between 2 & 3; and the least being 2}

iii) Thus from the above 3 equations, left side limit is not equal to right side limit.
So limit of the function does not exist.
Hence it is discontinuous at x = 2

So, In general, Greatest integer function f(x) is not continuous at all integral points.

Regards