Q. Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } at x = 1.
A.
Continuous
B.
Not continuous
C.
Depends on the limit
D.
Only left continuous
Show solution
Solution
The left limit as x approaches 1 is 1, the right limit is 2, and f(1) = 2. Since the left and right limits do not match, f(x) is not continuous at x = 1.
Correct Answer:
B
— Not continuous
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Q. Determine the continuity of the function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } at x = 1.
A.
Continuous
B.
Discontinuous
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 1, f(1) = 2(1) - 1 = 1 and lim x→1- f(x) = 1, lim x→1+ f(x) = 1. Thus, f(x) is continuous at x = 1.
Correct Answer:
A
— Continuous
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Q. Determine the continuity of the function f(x) = |x| at x = 0. (2020)
A.
Continuous
B.
Not continuous
C.
Depends on the limit
D.
Only left continuous
Show solution
Solution
The function f(x) = |x| is continuous at x = 0 since both the left-hand limit and right-hand limit equal f(0) = 0.
Correct Answer:
A
— Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)
A.
5, Continuous
B.
0, Not continuous
C.
5, Not continuous
D.
0, Continuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5, and the function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine its continuity.
A.
5, Continuous
B.
0, Continuous
C.
5, Not Continuous
D.
0, Not Continuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?
A.
5, Continuous
B.
5, Discontinuous
C.
0, Continuous
D.
0, Discontinuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0 if defined as f(0) = 5.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.
A.
1, Continuous
B.
0, Continuous
C.
1, Discontinuous
D.
0, Discontinuous
Show solution
Solution
The limit lim (x -> 0) (sin(x)/x) = 1. Since the limit exists and equals the function value at x = 0, it is continuous.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x). Is the function continuous at x = 0?
A.
1, Continuous
B.
0, Continuous
C.
1, Discontinuous
D.
0, Discontinuous
Show solution
Solution
The limit is 1, and if we define f(0) = 1, then f(x) is continuous at x = 0.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 3) (x^2 - 9)/(x - 3). Is the function continuous at x = 3? (2021)
A.
0, Yes
B.
0, No
C.
6, Yes
D.
6, No
Show solution
Solution
lim (x -> 3) (x^2 - 9)/(x - 3) = lim (x -> 3) (x + 3) = 6. The function is not defined at x = 3, hence not continuous.
Correct Answer:
C
— 6, Yes
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Q. Evaluate the limit lim x→2 (x^2 - 4)/(x - 2).
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Using L'Hôpital's Rule, lim x→2 (x^2 - 4)/(x - 2) = lim x→2 (2x)/(1) = 4.
Correct Answer:
C
— 4
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Q. For the function f(x) = x^3 - 3x + 2, find the points of discontinuity.
A.
None
B.
x = 1
C.
x = -1
D.
x = 2
Show solution
Solution
f(x) is a polynomial function and is continuous everywhere, hence no points of discontinuity.
Correct Answer:
A
— None
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Q. For the function f(x) = { 2x + 1, x < 1; 3, x = 1; x^2, x > 1 }, is f(x) continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
The left limit as x approaches 1 is 3, the right limit is 1, and f(1) = 3. Since the limits do not match, f(x) is discontinuous at x = 1.
Correct Answer:
B
— No
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Q. For the function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 }, is f(x) continuous at x = 0?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
The left limit as x approaches 0 is 0, the right limit is 1, and f(0) = 0. Since the limits do not match, f(x) is discontinuous at x = 0.
Correct Answer:
B
— No
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Q. For the function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 }, is f(x) continuous at x = 2?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 2, left limit is 4 and right limit is 4, but f(2) = 4. Hence, f(x) is continuous at x = 2.
Correct Answer:
B
— No
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Q. For the function f(x) = { x^2, x < 3; 9, x = 3; x + 3, x > 3 }, is f(x) continuous at x = 3?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
The left limit as x approaches 3 is 9, the right limit is also 9, and f(3) = 9. Therefore, f(x) is continuous at x = 3.
Correct Answer:
A
— Yes
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Q. For which value of c is the function f(x) = { x^2, x < 1; c, x = 1; 2x, x > 1 } continuous at x = 1? (2022)
Show solution
Solution
To make f(x) continuous at x = 1, we need c = 1^2 = 1. Thus, c must be 1.
Correct Answer:
B
— 2
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Q. For which value of k is the function f(x) = kx + 2 continuous at x = 3? (2023)
A.
k = 0
B.
k = 1
C.
k = -1
D.
k = 2
Show solution
Solution
The function f(x) = kx + 2 is a linear function and is continuous for all k at x = 3.
Correct Answer:
B
— k = 1
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Q. For which value of k is the function f(x) = { kx + 1, x < 2; 3, x >= 2 } continuous at x = 2?
Show solution
Solution
To be continuous at x = 2, k(2) + 1 must equal 3. Thus, 2k + 1 = 3, leading to k = 1.
Correct Answer:
B
— 2
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Q. For which value of k is the function f(x) = { kx + 1, x < 2; 3, x = 2; 2x - 1, x > 2 } continuous at x = 2?
Show solution
Solution
To ensure continuity at x = 2, k(2) + 1 must equal 3. Thus, k = 1.
Correct Answer:
B
— 2
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Q. For which value of k is the function f(x) = { kx + 1, x < 2; 3, x ≥ 2 } continuous at x = 2? (2019)
Show solution
Solution
To be continuous at x = 2, k(2) + 1 must equal 3. Thus, 2k + 1 = 3, giving k = 1.
Correct Answer:
B
— 2
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Q. If f(x) = 3x + 2, what is the value of f(1) and is it continuous?
A.
5, Continuous
B.
5, Not Continuous
C.
3, Continuous
D.
3, Not Continuous
Show solution
Solution
f(1) = 3(1) + 2 = 5. Since f(x) is a linear function, it is continuous everywhere.
Correct Answer:
A
— 5, Continuous
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Q. If f(x) = 3x + 2, what is the value of f(2) and is it continuous?
A.
8, Continuous
B.
8, Discontinuous
C.
7, Continuous
D.
7, Discontinuous
Show solution
Solution
f(2) = 3(2) + 2 = 8. Since f(x) is a polynomial, it is continuous everywhere.
Correct Answer:
A
— 8, Continuous
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Q. If f(x) = x^2 + 2x + 1, what is f(-1)? Is f(x) continuous at x = -1? (2019)
A.
0, Yes
B.
0, No
C.
1, Yes
D.
1, No
Show solution
Solution
f(-1) = (-1)^2 + 2*(-1) + 1 = 0. The function is a polynomial and is continuous everywhere, including at x = -1.
Correct Answer:
C
— 1, Yes
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Q. If f(x) = x^2 + 3x + 2, what is f(1) and is it continuous?
A.
6, Continuous
B.
6, Discontinuous
C.
5, Continuous
D.
5, Discontinuous
Show solution
Solution
f(1) = 1^2 + 3(1) + 2 = 6. Since f(x) is a polynomial function, it is continuous everywhere.
Correct Answer:
A
— 6, Continuous
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Q. If f(x) = x^2 + 3x + 2, what is the limit as x approaches -1?
Show solution
Solution
lim x→-1 f(x) = (-1)^2 + 3(-1) + 2 = 1 - 3 + 2 = 0.
Correct Answer:
C
— 2
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Q. If f(x) = x^2 + 3x + 2, what is the value of f(-1) and is it continuous?
A.
0, Continuous
B.
0, Discontinuous
C.
4, Continuous
D.
4, Discontinuous
Show solution
Solution
f(-1) = (-1)^2 + 3(-1) + 2 = 0. Since f(x) is a polynomial, it is continuous everywhere.
Correct Answer:
C
— 4, Continuous
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Q. If f(x) = x^2 - 4, what is the continuity of f(x) at x = 2?
A.
Continuous
B.
Not Continuous
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
f(x) = x^2 - 4 is a polynomial function, which is continuous everywhere, including at x = 2.
Correct Answer:
A
— Continuous
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Q. If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f(x) continuous at x = 1? (2019)
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 1, f(1) = 1^2 = 1 and the limit from the left is also 1, hence f(x) is continuous at x = 1.
Correct Answer:
A
— Yes
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Q. If f(x) = x^2 for x < 1 and f(x) = 3 for x ≥ 1, is f(x) continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 1, f(1) = 3 and limit from left is 1^2 = 1. Since they are not equal, f(x) is discontinuous at x = 1.
Correct Answer:
B
— No
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Q. If f(x) = x^3 - 3x + 2, what is f(1)? Is f(x) continuous at x = 1? (2019)
A.
0, Yes
B.
0, No
C.
1, Yes
D.
1, No
Show solution
Solution
f(1) = 1^3 - 3*1 + 2 = 0. The function is a polynomial and hence continuous everywhere, including at x = 1.
Correct Answer:
C
— 1, Yes
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Showing 1 to 30 of 54 (2 Pages)
Continuity MCQ & Objective Questions
Continuity is a fundamental concept in mathematics that plays a crucial role in various exams. Understanding this topic is essential for students aiming to excel in their school exams and competitive tests. Practicing MCQs and objective questions on continuity helps reinforce concepts, making it easier to tackle important questions during exams. Regular practice not only boosts confidence but also enhances problem-solving skills, leading to better scores.
What You Will Practise Here
Definition and properties of continuity
Types of continuity: pointwise and uniform
Continuity of functions: polynomial, rational, and trigonometric
Intermediate Value Theorem and its applications
Limits and their relationship with continuity
Graphical interpretation of continuous functions
Common continuity problems and their solutions
Exam Relevance
The topic of continuity is frequently tested in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that assess their understanding of the definitions, properties, and applications of continuity. Common question patterns include identifying continuous functions from graphs, applying the Intermediate Value Theorem, and solving problems that require determining the continuity of given functions. Mastering this topic is essential for achieving high marks in mathematics.
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 piecewise functions
Failing to apply the Intermediate Value Theorem correctly
FAQs
Question: What is continuity in mathematics?Answer: Continuity refers to a function being unbroken and having no gaps, jumps, or holes in its graph over a given interval.
Question: How can I improve my understanding of continuity?Answer: Regular practice of continuity MCQ questions and reviewing key concepts will help solidify your understanding and prepare you for exams.
Don't wait any longer! Start solving practice MCQs on continuity today to test your understanding and boost your exam preparation. Your success is just a question away!
