Q. The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Correct Answer:
B
— No
Learn More →
Q. The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:
A.
Continuous at x = 0
B.
Not continuous at x = 0
C.
Continuous everywhere
D.
None of the above
Show solution
Solution
The function is not continuous at x = 0 since the limit does not equal f(0).
Correct Answer:
B
— Not continuous at x = 0
Learn More →
Q. The function f(x) = { 2x + 3, x < 1; x^2 + 1, x >= 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.
Correct Answer:
B
— 1
Learn More →
Q. The function f(x) = { 3x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1 if which condition holds?
A.
3 = 2
B.
1 = 2
C.
2 = 1
D.
2 = 4
Show solution
Solution
For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.
Correct Answer:
A
— 3 = 2
Learn More →
Q. The function f(x) = { 3x + 1, x < 1; 2x + 3, x >= 1 } is continuous at x = 1 if:
Show solution
Solution
For continuity at x = 1, both pieces must equal 4, hence the function is continuous.
Correct Answer:
A
— 3
Learn More →
Q. The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
Learn More →
Q. The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Correct Answer:
A
— Yes
Learn More →
Q. The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?
A.
x = -1
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.
Correct Answer:
B
— x = 0
Learn More →
Q. The function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } is:
A.
Continuous everywhere
B.
Continuous at x = 1
C.
Not continuous at x = 1
D.
Continuous for x < 1
Show solution
Solution
The function is not continuous at x = 1 because the left-hand limit does not equal the function value.
Correct Answer:
C
— Not continuous at x = 1
Learn More →
Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at which point?
A.
x = 0
B.
x = 1
C.
x = 2
D.
x = -1
Show solution
Solution
To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.
Correct Answer:
B
— x = 1
Learn More →
Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
Learn More →
Q. The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:
A.
f(2) = 4
B.
lim x->2 f(x) = 4
C.
Both a and b
D.
None of the above
Show solution
Solution
Both conditions must hold true for continuity at x = 2.
Correct Answer:
C
— Both a and b
Learn More →
Q. The function f(x) = { x^2, x < 2; k, x = 2; 3x - 4, x > 2 } is continuous at x = 2 for which value of k?
Show solution
Solution
To be continuous at x = 2, k must equal f(2) = 2^2 = 4.
Correct Answer:
C
— 4
Learn More →
Q. What is the limit of f(x) = 1/x as x approaches 0 from the right?
A.
0
B.
Infinity
C.
1
D.
Does not exist
Show solution
Solution
As x approaches 0 from the right, f(x) approaches infinity, indicating a discontinuity at x = 0.
Correct Answer:
B
— Infinity
Learn More →
Q. What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?
Show solution
Solution
Setting 2k + 2 = 0 gives k = 2.
Correct Answer:
C
— 2
Learn More →
Q. What is the value of k for which the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0?
Show solution
Solution
Setting k(0) = 0^2 + 1 gives k = 1.
Correct Answer:
B
— 0
Learn More →
Q. What is the value of k for which the function f(x) = { kx, x < 2; x^2, x >= 2 } is continuous at x = 2?
Show solution
Solution
Setting k(2) = 2^2 gives 2k = 4, thus k = 2.
Correct Answer:
C
— 4
Learn More →
Q. What is the value of p for which the function f(x) = { 3x + p, x < 2; x^2 - 4, x >= 2 } is continuous at x = 2?
Show solution
Solution
Setting the two pieces equal at x = 2: 3(2) + p = 2^2 - 4. Solving gives p = -2.
Correct Answer:
A
— -1
Learn More →
Q. What is the value of q for which the function f(x) = { 5 - q, x < 1; 3x + 2, x >= 1 } is continuous at x = 1?
Show solution
Solution
Setting the two pieces equal at x = 1: 5 - q = 3(1) + 2. Solving gives q = 0.
Correct Answer:
C
— 2
Learn More →
Q. What value of a makes the function f(x) = { 2x + 1, x < 1; a, x = 1; x^2 + 1, x > 1 continuous at x = 1?
Show solution
Solution
Setting 2(1) + 1 = a and a = 2 for continuity.
Correct Answer:
B
— 2
Learn More →
Q. What value of a makes the function f(x) = { 2x + a, x < 3; 5, x = 3; x^2 - 1, x > 3 continuous at x = 3?
Show solution
Solution
Setting 2(3) + a = 5 gives a = -1.
Correct Answer:
C
— 2
Learn More →
Q. What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?
Show solution
Solution
Setting 4 = 2 gives a = 1 for continuity.
Correct Answer:
B
— 0
Learn More →
Q. What value of k makes the function f(x) = { kx, x < 1; 2, x = 1; x + 1, x > 1 continuous at x = 1?
Show solution
Solution
Setting the left limit (k(1) = k) equal to the right limit (1 + 1 = 2), we find k = 2.
Correct Answer:
B
— 1
Learn More →
Q. What value of m makes the function f(x) = { 3x + 1, x < 2; mx + 4, x = 2; x^2 - 1, x > 2 continuous at x = 2?
Show solution
Solution
Setting the left limit (3(2) + 1 = 7) equal to the right limit (2^2 - 1 = 3), we find m = 3.
Correct Answer:
D
— 4
Learn More →
Q. Which of the following functions is continuous at x = 2?
A.
f(x) = 1/x
B.
f(x) = x^2 - 4
C.
f(x) = sin(1/x)
D.
f(x) =
.
x
.
Show solution
Solution
f(x) = x^2 - 4 is a polynomial function and is continuous everywhere, including at x = 2.
Correct Answer:
B
— f(x) = x^2 - 4
Learn More →
Q. Which of the following functions is continuous at x = 2? f(x) = { x^2 - 4, x < 2; 3x - 6, x >= 2 }
A.
Continuous
B.
Not continuous
C.
Depends on k
D.
None of the above
Show solution
Solution
At x = 2, f(2) = 0 and limit from left is 0, limit from right is also 0. Hence, it is continuous.
Correct Answer:
A
— Continuous
Learn More →
Q. Which of the following functions is continuous at x = 2? f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }
A.
Continuous
B.
Not continuous
C.
Depends on k
D.
None of the above
Show solution
Solution
To check continuity at x = 2, we find the left limit (4), right limit (4), and f(2) (4). All are equal, so f(x) is continuous.
Correct Answer:
A
— Continuous
Learn More →
Q. Which of the following functions is continuous everywhere?
A.
f(x) = 1/x
B.
f(x) = x^2
C.
f(x) = sin(x)
D.
f(x) =
.
x
.
Show solution
Solution
f(x) = x^2 is a polynomial function and is continuous everywhere.
Correct Answer:
B
— f(x) = x^2
Learn More →
Q. Which of the following functions is not continuous at x = 0?
A.
f(x) = x^3
B.
f(x) = e^x
C.
f(x) = 1/x
D.
f(x) = ln(x)
Show solution
Solution
The function f(x) = 1/x is not defined at x = 0, hence it is not continuous there.
Correct Answer:
C
— f(x) = 1/x
Learn More →
Q. Which of the following functions is not continuous at x = 1?
A.
f(x) = x^2
B.
f(x) = 1/x
C.
f(x) = sin(x)
D.
f(x) = { x, x < 1; 2, x >= 1 }
Show solution
Solution
The function has a jump discontinuity at x = 1, hence it is not continuous.
Correct Answer:
D
— f(x) = { x, x < 1; 2, x >= 1 }
Learn More →
Showing 91 to 120 of 124 (5 Pages)
Continuity MCQ & Objective Questions
Understanding the concept of "Continuity" is crucial for students preparing for school exams and competitive tests in India. Mastering this topic not only enhances your conceptual clarity but also boosts your confidence in tackling objective questions. Practicing MCQs and important questions on Continuity can significantly improve your exam performance, making it essential for effective exam preparation.
What You Will Practise Here
Definition and properties of continuity
Types of continuity: point continuity and interval continuity
Continuity of functions and their graphical representations
Intermediate Value Theorem and its applications
Limits and their role in establishing continuity
Common functions that exhibit continuity
Real-life applications of continuous functions
Exam Relevance
The topic of Continuity is frequently featured in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect questions that assess their understanding of continuity through MCQs that may involve identifying continuous functions, applying the Intermediate Value Theorem, or solving problems related to limits. Familiarity with common question patterns will help you tackle these effectively.
Common Mistakes Students Make
Confusing continuity with differentiability
Overlooking the importance of limits in determining continuity
Misinterpreting graphical representations of continuous functions
Neglecting to check endpoints in interval continuity
Failing to apply the Intermediate Value Theorem correctly
FAQs
Question: What is the definition of continuity in mathematics?Answer: Continuity refers to a function being unbroken or uninterrupted over an interval, meaning small changes in input result in small changes in output.
Question: How can I determine if a function is continuous at a point?Answer: A function is continuous at a point if the limit exists at that point, the function is defined at that point, and the limit equals the function's value.
Start solving practice MCQs on Continuity today to enhance your understanding and prepare effectively for your exams. Remember, consistent practice is key to mastering this important topic!
