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Continuity and Discontinuity of Function| Calculus Math

Continuity and Discontinuity of Function| Calculus Math

05/16/2021
Continuity and Discontinuity of Function| Calculus Math (Definition and Examples with Q & N)
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Continuity and discontinuity in calculus 

Introduction and basic concept

In this section, we shall study the discontinuity of the given different functions. And we will also study the types of discontinuity also. Which are discussed below.

Left-hand limit

A left-hand limit means the limit of a function as it approaches from the left-hand side. Hence left-hand limit can be defined as the function f(x) as the value of the x approaches from the left. It is denoted as

Right-hand limit

A right-hand limit means the limit of a function as it approaches from the right-hand side. Hence right-hand limit can be defined as the function f(x) as the value of the x approaches from the right. It is denoted as 

Method of finding the left hand and right-hand limit

To find the left hand as well as right-hand limit we need to replace x by (x+h) and take the limit as h→0.

Limit of a function

Any limit   is said to be existed only when left and limit and right-hand limit are equal i.e.

.

Method of distinguishing left-hand limit and right-hand limit.

more often we confuse which part of the problem is the left-hand limit and which part of the problem is the right-hand limit. Here is a very easy way to distinguish left and limit and right-hand limit. 

  • if there is X>... (something) then take it as right-hand limit as . + Sign denote Righ hand limit. 
  • if there is X<...(something ) then take it as left-hand limit as . - Sign denote left hand limit.

 

The simple rule is that If x> something then take it as positive i.e, right-hand limit and if x< something then takes it as negative i.e, left-hand limit. + and - sign has no role in algebric function but it has a role in trigonometric and absolute function. 

(Note: generally, we can directly put LHL and RHL values irrespective of sign, but we need to sign in trigonometric function and absolute value which are done in the solved example of this book.)

 

Continuity: Defintion and approches in calculus

any function f(x) is said to be continuous at any point ‘a’ of its domain if given any positive number σ there is corresponding a positive number δ such that

 

For all the values of h such that

.

In another form,

A real function f(x) is said to be continuous at a point an of its domain if   and equal to f(a) i.e.

.

i.e. function will be continuous if left hand limit, right-hand limit, and functional value of f(x) at x=a are equal.

right hand limit=left hand limit= functional value

Three Conditions of Continuity

To be a continuous function, the following conditions should be fulfilled.

1. left-hand limit and right-hand limit should be equal i.e.   must exist and 

 i.e left-hand limit and right-hand limit are quals.

 

2. There exist function value i.e. f(a) exist.

 

3. left-hand limit, right-hand limit and functional value are equal. 

 

Discontinue Function

Lack of continuity is defined as discontinuity. in mathematics it is the point at which a mathematical object is discontinuous. Okay, let's have some literal meaning of the continuity and discontinuity. Actually, what is the mean of continuity and discontinuity? if I say Edubomb.com website server is running for week continuisly, what does it means? it means the server is working without stopping or breaking. 

Similarly, in mathematics, any function is said to be a continuous Function either at point or interval if the graph of such function passes without interrupting or breaking. If there is a break or interruption in the graph then it is said to be discontinued. 

Types of discontinuity

1. Infinite discontinuity (with graph)

In this discontinuity, the function diverges at x=a so that discontinuity nature is observed. At this time value of the function tend to infinite i.e 

  

 this shows that function f(x) is discontinues at x = a.

The following graph shows the infinite discontinuity.

Infinite discontinuity examples:

when both sided limit (Left-hand limit as well as right-hand limit) tends to infinite then it is said to be infinite discontinuity. 

let us consider the function 

 

when we put x=2 we get 

When we take the left-hand limit and right-hand limit we get infinite value.

infinite discontinuity

2. Ordinary discontinuity or jump discontinuity (with graph)

This discontinuity arises when there exists both left-hand limits as well as the right hand but they are not equal. I.e.

 hence  also does not exist. This is shown in the following figures.

Ordinary discontinuity

Example of Ordinary discontinuity or jump discontinuity

when both left-hand and right-hand limit exist but they are not equal then it is called the ordinary discontinuity. For example 

continuity of the function f(x) at  x=0

  

is ordinary discontinuity or jump discontinuity. 

Because:

 

We have two functions,

Left-hand limit at x=0

 

=(2.0-1)=-1 

Right hand limit at x = 0

li 

=(2.0+1)=1 

LHL≠RHL

3. Removable discontinuity (with graph)

Such type of discontinuity arises when there is a well-defined two-sided limit but there is no well-defined function value i.e. f(a) or this f(a) is not equal to the left-hand limit and right-hand limit. I.e.

 at x=a. such discontinuity can be removed by redefining the given function.  This is shown in the following figure.

Removable discontinuity

Example of Removable discontinuity

in this case functional value does not exist indefinable form. i.e. 

for examples 

This function has removable discontinuity.

Because:

Right-hand limit at x=2

 

=2-4=-2 

Left-hand limit at x=2

 

 

Functional value at x=2

f(2)=3 

since LHL= RHL≠ functional value at x=2

 

Copy and Paste are restricted 

SOLVED EXAMPLE

Example-1 show that given function f(x)=x3 is continuous at x=a ?

Solution:

We have given function f(x)=x3

Left-hand limit at x=a

 

Right-hand limit at x = a

  

Functoonal vlaue of f(x) at x=a   

F(x)=X3=23=8 

Hence from the above left-hand limit, right-hand limit, and functional values all are equal i.e.

LHL=RHL=F.V.

All three conditions are fulfilled 

Hence given function is continue at x=a.

Example-2 discuss the continuity of the function f(x) at  x=0

  

Solution:

We have two functions,

Left-hand limit at x=0

 

=(2.0-1)=-1 

Right hand limit at x = 0

li 

=(2.0+1)=1 

 

Functonal vlaue of f(x) at x=0   

f(0)=2x+1=2×0+1= 0+1=0

Since LHL≠RHL hence given function cannot be continuous at x=0.

Hence from above left-hand limit, right-hand limit, and functional values all are not equal i.e.

LHLRHL 

Hence given function is not continue at x=0.

 

Example-3 discuss the continuity of the function

  

Solution:

We have two functions,

Left-hand limit at x0

  

 

 

Right-hand limit at x ≠ a

 

 

 

Functonal vlaue of f(x) at x=0   

  

Since LHL≠RHL hence given function cannot be continuous.

Hence from the above left-hand limit, right-hand limit, and functional values all are equal i.e.

LHLRHL 

Hence given function is not continuous.

 

Example-4. Show that the function f(x)=2x-|x| is continuous at x=0.

Solution:

 We have

Left-hand limit at x=0

 

=2(-0)-|-0|=0 

Right-hand limit at x=0

 

=2(0)-|0|=0 

 Functional value at x=0

 f(x)=2x-|x|=2.0-|0|=0

Since LHL=RHL=FV

Hence given function is continuous at x=0.

 

Example-5.  If the given function is continuous at x=0. Find the value of k.

 

Solution:

Right-hand limit at x≠ 0.

 

 

 

left-hand limit at x≠ 0.

 

 

 

Since the given function is continuous hence, we must have that left-hand limit and right-hand limit and function value must be equal i.e.

LHL=RHL=FV=1

Or functional value f(x)=1

But given that f(x)=1

Hence k=1.

Example:6 Let a function f(x) be defined by

 

Verify that the limit of the function that exists at x=2 is a function continuous at x=2? If not, how can you make it continuous?

Solution:

(note: of x> something then take is the as positive and right-hand limit and if x< something then takes it as a negative and left-hand limit.)

Right-hand limit at x=2

 

=2-4=-2 

Left-hand limit at x=2

 

 

Functional value at x=2

f(2)=3 

since LHL= RHL≠ functional value at x=2 hence given function is not continuous at x=2.

We can make given function continuous when we put f(x)=-2. And redefining as 

 

Q and A 

Q. How do you find the continuity of function?

Answer:

To be a continuous function, the following conditions should be fulfilled.

1. left-hand limit and right-hand limit should be equal i.e.   must exist and 

 i.e left-hand limit and right-hand limit are quals.

 

2. There exist function value i.e. f(a) exist.

 

3. left-hand limit, right-hand limit and functional value are equal. 

Q. What is the meaning of continuity of function?

Answer:

 

Any function is said to be continuous if the graph of the given function pass without break or interferes. any function f(x) is said to be continuous at any point ‘a’ of its domain if given any positive number σ there is corresponding a positive number δ such that

 

For all the values of h such that

.

Q. what is the difference between continuity and discontinuity?

Answers:

Lack of continuity is defined as discontinuity. in mathematics it is the point at which a mathematical object is discontinuous. Okay, let's have some literal meaning of the continuity and discontinuity. Actually, what is the mean of continuity and discontinuity? if I say the Edubomb.com website server is running for a week continuously, what does it means? it means the server is working without stopping or breaking. 

Similarly, in mathematics, any function is said to be a continuous Function either at point or interval if the graph of such function passes without interrupting or breaking. If there is a break or interruption in the graph then it is said to be discontinued. 

Difference between continuity and discontinuity

Continuity  Discontinuity 
1. Graph if given function pass without breaks  1. Graph of given function pass with breaks
2. left-hand limit, right-hand limit, and functional value are equal  2. left-hand limit, right-hand limit, and functional value are not equal.

 

EXERCISE 

1. What is the meaning of continuity and discontinuity in mathematics? define in detail.

2. what are three conditions for continuity in mathematics?

3. what are three types of discontinuity? define them with figures.

4. How do you find the continuity of a given function?

5. write any three differences between continuity and discontinuity?

6. what are the examples of continuity and discontinuity in real life?

 

Want to study more topics?

 





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