Question

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.

Solution

Let f(x) = [ x ]. D_f = R
Let a be any real number ∈ D_f.
Two cases arise:
Case I. If a is not an integer, then
$Lt with straight x rightwards arrow straight a below space straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight a below space left square bracket straight x right square bracket equals left square bracket straight a right square bracket equals straight f left parenthesis straight a right parenthesis$
⇒ f is continuous at x = a

Case II. If a ∈ 1, then f(a) = [ a ] = a and
$Lt with straight x rightwards arrow straight a to the power of minus below space straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight a to the power of minus below left square bracket straight x right square bracket equals straight a minus 1 Lt with straight x rightwards arrow straight a to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight a to the power of plus below space left square bracket straight x right square bracket equals straight a therefore Lt with straight x rightwards arrow straight a to the power of minus below straight f left parenthesis straight x right parenthesis space not equal to Lt with straight x rightwards arrow straight a to the power of plus below space straight f left parenthesis straight x right parenthesis$
∴ f is not continuous at x = a, a ∈ I.
∴ function is discontinuous at every integral point.

Tips: -

1. Domain of continuity for the function [x] is R – I.
2. From the graph of [x], done in earlier class, it is also clear that [x] is discontinuous at integral points.

For Daily Free Study Material Join wiredfaculty

Continuity And Differentiability

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.

Some More Questions From Continuity and Differentiability Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location