HOW TO FIND POINTS OF DISCONTINUITY FOR A PIECEWISE FUNCTION

Example 1 :

Find the points of discontinuity of the function f, where

Solution :

For the values of x greater than 3, we have to select the function 4x + 5.

lim x->3- f(x)  =  lim x->3- 4x + 5

  =  4(3) + 5 

  =  12 + 5

  =  17  -------(1)

For the values of x lesser than 3, we have to select the function 4x - 5.

lim x->3+ f(x)  =  lim x->3+ 4x - 5

  =  4(3) - 5 

  =  12 - 5

  =  7  -------(2)

lim x->3- f(x) ≠ lim x->3+ f(x)

So, the given piece-wise function is not continuous at

x  =  3

Example 2 :

Find the points of discontinuity of the function f, where

Solution :

For the values of x greater than 2, we have to select the function x + 2.

lim x->2- f(x)  =  lim x->2- x + 2

  =  2 + 2

       =  4  -------(1)

For the values of x lesser than 2, we have to select the function x2.

lim x->2+ f(x)  =  lim x->2+ x2

  =  22

       =  4-------(2)

lim x->2- f(x) =  lim x->2+ f(x)

The function is continuous at x = 2.

So, the given piece-wise function is continuous for all real values of x. 

That is, 

∈ R

Example 3 :

Find the points of discontinuity of the function f, where

Solution :

For the values of x greater than 2, we have to select the function x2 + 1

lim x->2- f(x)  =  lim x->2- x2 + 1

  =   22 + 1

       =  5  -------(1)

For the values of x lesser than 2, we have to select the function x3 - 3.

lim x->2+ f(x)  =  lim x->2+ x- 3

  =  2- 3

=  8 - 3

       =  5-------(2)

lim x->2- f(x) =  lim x->2+ f(x)

The function is continuous at x  =  2.

So, the given piece-wise function is continuous for all real values of x. 

That is,

∈ R

Example 4 :

Find the points of discontinuity of the function f, where

Solution :

Here we are going to check the continuity between 0 and π/2.

For the values of x lesser than or equal to π/4, we have to choose the function sin x.

lim x->π/4- f(x)  =  lim x->π/4- sin x

       =  sin (π/4)

=  1/√2

For the values of x greater than π/4, we have to choose the function cos x .

lim x->π/4+ f(x)  =  lim x->π/4+ cos x

       =  cos (π/4)

=  1/√2

The function is continuous for all x ∈ [0, π/2).

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Videos

    May 22, 24 06:32 AM

    sattriangle1.png
    SAT Math Videos (Part 1 - No Calculator)

    Read More

  2. Simplifying Algebraic Expressions with Fractional Coefficients

    May 17, 24 08:12 AM

    Simplifying Algebraic Expressions with Fractional Coefficients

    Read More

  3. The Mean Value Theorem Worksheet

    May 14, 24 08:53 AM

    tutoring.png
    The Mean Value Theorem Worksheet

    Read More