Math24.net
Calculus
Home
Calculus
Limits and Continuity
Differentiation
Integration
Sequences and Series
Double Integrals
Triple Integrals
Line Integrals
Surface Integrals
Fourier Series
Differential Equations
1st Order Equations
2nd Order Equations
Nth Order Equations
Systems of Equations
Formulas and Tables
   Discontinuous Functions
If f (x) is not continuous at x = a, then f (x) is said to be discontinuous at this point. Figure 1 shows the graphs of four functions, two of which are continuous at x = a and two are not.
Function continuous at x=a
Function discontinuous at x=a
Continuous at x = a.
Discontinuous at x = a.
Function continuous at x=a
Function discontinuous at x=a
Continuous at x = a.
Discontinuous at x = a.
Figure 1.
Classification of Discontinuity Points
All discontinuity points are divided into discontinuities of the first and second kind.

The function f (x) has a discontinuity of the first kind at x = a if
  • There exist left-hand limit and right-hand limit ;
  • These one-sided limits are finite.
Further there may be the following two options:
  • The right-hand limit and the left-hand limit are equal to each other:
    Such a point is called a removable discontinuity.
  • The right-hand limit and the left-hand limit are unequal:
    In this case the function f (x) has a jump discontinuity.
The function f (x) is said to have a discontinuity of the second kind (or a nonremovable or essential discontinuity) at x = a, if at least one of the one-sided limits either does not exist or is infinite.

   Example 1
Investigate continuity of the function .

Solution.
The given function is not defined at x = −1 and x = 1. Hence, this function has discontinuities at x = ±1. To determine the type of the discontinuities, we find the one-sided limits:
     
Since the left-side limit at x = −1 is infinity, we have an essential discontinuity at this point.
     
Similarly, the right-side limit at x = 1 is infinity. Hence, here we also have an essential discontinuity.

   Example 2
Show that the function has a removable discontinuity at x = 0.

Solution.
Obviously, the function is not defined at x = 0. Since sin x is continuous at every x, then the initial function is also continuous for all x except the point x = 0.
Since , the function has a removable discontinuity at this point. We can construct the new function
     
which is continuous at every real x.

   Example 3
Find the points of discontinuity of the function if they exist.

Solution.
The function exists for all x, however it is defined by two different functions and, therefore, is not elementary. We investigate "behavior" of the function near to the point x = 0 where its analytic expression changes.

Calculate one-sided limits at x = 0.
     
Thus, the function has a discontinuity of the first kind at x = 0. The finite jump at this point is
     
The function is continuous for all other x, because both the functions defined from the left and from the right of the point x = 0 are elementary functions without any discontinuities.

   Example 4
Find the points of discontinuity of the function if they exist.

Solution.
This elementary function is defined for all x except x = 0, where it has a discontinuity. Find one-sided limits at
     
As can be seen, the function has a discontinuity point of the first kind at x = 0 (Figure 2).
graph of function arctan(1/x)
discontinuity of the first kind
Fig.2
Fig.3
   Example 5
Find the points of discontinuity of the function if they exist.

Solution.
The function is defined and continuous for all x except , where it has a discontinuity. Investigate this discontinuity point:
     
Since the values of the one-sided limits are finite, then there's a discontinuity of the first kind at the point . The graph of the function is sketched on Figure 3.

All Rights Reserved © www.math24.net, 2010-2014   info@math24.net