RD Sharma Class 12 Solutions - Continuity | Mathematics (Maths) Class 12 - JEE PDF Download

 Page 1


9. Continuity
Exercise 9.1
1. Question
Test the continuity of the following function at the origin :
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighbourhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
2. The idea of modulus function |x |: You can think this function as a machine in which you can give it any
real no. as an input and it returns its absolute value i.e. if positive is entered it returns the same no and if
negative is entered it returns the corresponding positive no.
Eg:– |2| = 2 ; |–2| = –(–2) = 2
Similarly, we can define it for variable x, if x = 0 |x| = x
If x < 0 |x| = (–x)
? 
Now we are ready to solve the question –
We need to check the continuity at the origin (0,0) i.e. we will check it at x=0.
So, we need to see whether at x=0,
IF, LHL = RHL = f(0)
i.e. 
For this question c = 0
Page 2


9. Continuity
Exercise 9.1
1. Question
Test the continuity of the following function at the origin :
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighbourhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
2. The idea of modulus function |x |: You can think this function as a machine in which you can give it any
real no. as an input and it returns its absolute value i.e. if positive is entered it returns the same no and if
negative is entered it returns the corresponding positive no.
Eg:– |2| = 2 ; |–2| = –(–2) = 2
Similarly, we can define it for variable x, if x = 0 |x| = x
If x < 0 |x| = (–x)
? 
Now we are ready to solve the question –
We need to check the continuity at the origin (0,0) i.e. we will check it at x=0.
So, we need to see whether at x=0,
IF, LHL = RHL = f(0)
i.e. 
For this question c = 0
f(x) can be rewritten using the concept of modulus function as :
 …… Equation 1
NOTE : 
Now we have three different expressions for different conditions of x.
LHL =  =  =  using eqn 1
RHL =  using eqn 1
LHL ? RHL so we even don’t need to check for f(0)
? We can easily say that f(x) is discontinuous at the origin.
2. Question
A function f(x) is defined as  Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds true, function is continuous else
it is discontinuous.
Let’s solve :
To prove function is continuous at x=3 we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3, therefore c=3. (in equation 1)
 ……Eqn 2
? f(3) = 5 using eqn 2
Page 3


9. Continuity
Exercise 9.1
1. Question
Test the continuity of the following function at the origin :
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighbourhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
2. The idea of modulus function |x |: You can think this function as a machine in which you can give it any
real no. as an input and it returns its absolute value i.e. if positive is entered it returns the same no and if
negative is entered it returns the corresponding positive no.
Eg:– |2| = 2 ; |–2| = –(–2) = 2
Similarly, we can define it for variable x, if x = 0 |x| = x
If x < 0 |x| = (–x)
? 
Now we are ready to solve the question –
We need to check the continuity at the origin (0,0) i.e. we will check it at x=0.
So, we need to see whether at x=0,
IF, LHL = RHL = f(0)
i.e. 
For this question c = 0
f(x) can be rewritten using the concept of modulus function as :
 …… Equation 1
NOTE : 
Now we have three different expressions for different conditions of x.
LHL =  =  =  using eqn 1
RHL =  using eqn 1
LHL ? RHL so we even don’t need to check for f(0)
? We can easily say that f(x) is discontinuous at the origin.
2. Question
A function f(x) is defined as  Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds true, function is continuous else
it is discontinuous.
Let’s solve :
To prove function is continuous at x=3 we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3, therefore c=3. (in equation 1)
 ……Eqn 2
? f(3) = 5 using eqn 2
LHL = 
Using equation 2 –
RHL = 
Clearly, LHL = RHL = f(3) = 5
? f(x) is continuous at x=3
3. Question
A function f(x) is defined as 
Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
Let’s solve :
To prove function is continuous at x=3, we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3 therefore c=3 (in equation 1)
Page 4


9. Continuity
Exercise 9.1
1. Question
Test the continuity of the following function at the origin :
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighbourhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
2. The idea of modulus function |x |: You can think this function as a machine in which you can give it any
real no. as an input and it returns its absolute value i.e. if positive is entered it returns the same no and if
negative is entered it returns the corresponding positive no.
Eg:– |2| = 2 ; |–2| = –(–2) = 2
Similarly, we can define it for variable x, if x = 0 |x| = x
If x < 0 |x| = (–x)
? 
Now we are ready to solve the question –
We need to check the continuity at the origin (0,0) i.e. we will check it at x=0.
So, we need to see whether at x=0,
IF, LHL = RHL = f(0)
i.e. 
For this question c = 0
f(x) can be rewritten using the concept of modulus function as :
 …… Equation 1
NOTE : 
Now we have three different expressions for different conditions of x.
LHL =  =  =  using eqn 1
RHL =  using eqn 1
LHL ? RHL so we even don’t need to check for f(0)
? We can easily say that f(x) is discontinuous at the origin.
2. Question
A function f(x) is defined as  Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds true, function is continuous else
it is discontinuous.
Let’s solve :
To prove function is continuous at x=3 we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3, therefore c=3. (in equation 1)
 ……Eqn 2
? f(3) = 5 using eqn 2
LHL = 
Using equation 2 –
RHL = 
Clearly, LHL = RHL = f(3) = 5
? f(x) is continuous at x=3
3. Question
A function f(x) is defined as 
Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
Let’s solve :
To prove function is continuous at x=3, we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3 therefore c=3 (in equation 1)
 …… eqn 2
From eqn 2 :
f(3) = 6
LHL = 
Using equation 2 –
RHL = 
Clearly, LHL = RHL = f(3) = 6
? f(x) is continuous at x=3
4. Question
If  Find whether f(x) is continuous at x = 1.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
Let’s solve :
To check whether function is continuous at x=3 we need to check whether LHL = RHL = f(c)
As continuity is to be checked at x = 1 therefore c=1 (in equation 1)
Page 5


9. Continuity
Exercise 9.1
1. Question
Test the continuity of the following function at the origin :
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighbourhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
2. The idea of modulus function |x |: You can think this function as a machine in which you can give it any
real no. as an input and it returns its absolute value i.e. if positive is entered it returns the same no and if
negative is entered it returns the corresponding positive no.
Eg:– |2| = 2 ; |–2| = –(–2) = 2
Similarly, we can define it for variable x, if x = 0 |x| = x
If x < 0 |x| = (–x)
? 
Now we are ready to solve the question –
We need to check the continuity at the origin (0,0) i.e. we will check it at x=0.
So, we need to see whether at x=0,
IF, LHL = RHL = f(0)
i.e. 
For this question c = 0
f(x) can be rewritten using the concept of modulus function as :
 …… Equation 1
NOTE : 
Now we have three different expressions for different conditions of x.
LHL =  =  =  using eqn 1
RHL =  using eqn 1
LHL ? RHL so we even don’t need to check for f(0)
? We can easily say that f(x) is discontinuous at the origin.
2. Question
A function f(x) is defined as  Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds true, function is continuous else
it is discontinuous.
Let’s solve :
To prove function is continuous at x=3 we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3, therefore c=3. (in equation 1)
 ……Eqn 2
? f(3) = 5 using eqn 2
LHL = 
Using equation 2 –
RHL = 
Clearly, LHL = RHL = f(3) = 5
? f(x) is continuous at x=3
3. Question
A function f(x) is defined as 
Show that f(x) is continuous at x = 3.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighborhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighborhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
Let’s solve :
To prove function is continuous at x=3, we need to show LHL = RHL = f(c) As continuity is to be checked at x
= 3 therefore c=3 (in equation 1)
 …… eqn 2
From eqn 2 :
f(3) = 6
LHL = 
Using equation 2 –
RHL = 
Clearly, LHL = RHL = f(3) = 6
? f(x) is continuous at x=3
4. Question
If  Find whether f(x) is continuous at x = 1.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.
Let’s solve :
To check whether function is continuous at x=3 we need to check whether LHL = RHL = f(c)
As continuity is to be checked at x = 1 therefore c=1 (in equation 1)
As,  ………eqn 2
From eqn 2 :
f(1) = 2
LHL = 
Using equation 2 –
RHL = 
Clearly, LHL = RHL = f(1) = 2
? f(x) is continuous at x=1
5. Question
If  Find whether f(x) is continuous at x = 0.
Answer
Ideas required to solve the problem:
1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x), we
can say that if we plot the coordinates (x, f(x)) and try to join all those points in the specified region, we can
do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve
without any breakage.
Mathematically we define the same thing as given below:
A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to
be checked
If:–
 equation 1
where h is a very small positive no (can assume h = 0.00000000001 like this )
It means:–
Limiting the value of the left neighbourhood of x = c also called left–hand limit LHL  must be
equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both
must be equal to the value of f(x) at x=c i.e. f(c).
Thus, it is the necessary condition for a function to be continuous
So, whenever we check continuity we try to check above equality if it holds, function is continuous else it is
discontinuous.