Q. If f(x) = x^3 - 3x + 2, what is the value of f(1) and is it continuous?
A.
0, Continuous
B.
0, Not Continuous
C.
1, Continuous
D.
1, Not Continuous
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Solution
f(1) = 1^3 - 3(1) + 2 = 0. Since f(x) is a polynomial, it is continuous everywhere.
Correct Answer:
A
— 0, Continuous
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Q. Is the function f(x) = 1/(x-1) continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
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Solution
The function f(x) = 1/(x-1) is not defined at x = 1, hence it is discontinuous at that point.
Correct Answer:
B
— No
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Q. Is the function f(x) = sqrt(x) continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
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Solution
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = |x| continuous at x = 0?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
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Solution
The function f(x) = |x| is continuous at x = 0 because the left limit, right limit, and f(0) all equal 0.
Correct Answer:
A
— Yes
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Q. The function f(x) = 1/(x-1) is continuous on which of the following intervals?
A.
(-∞, 1)
B.
(1, ∞)
C.
(-∞, ∞)
D.
(-∞, 0)
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Solution
The function f(x) = 1/(x-1) is discontinuous at x = 1, hence it is continuous on (1, ∞).
Correct Answer:
B
— (1, ∞)
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Q. The function f(x) = 2x + 1 is continuous at which of the following intervals?
A.
(-∞, ∞)
B.
(0, 1)
C.
(1, 2)
D.
(2, 3)
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Solution
f(x) = 2x + 1 is a linear function and is continuous over the entire real line (-∞, ∞).
Correct Answer:
A
— (-∞, ∞)
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Q. The function f(x) = 2x + 3 is continuous at which of the following intervals?
A.
(-∞, ∞)
B.
[0, 1]
C.
[1, 2]
D.
[2, 3]
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Solution
f(x) = 2x + 3 is a linear function and is continuous over the entire real line (-∞, ∞).
Correct Answer:
A
— (-∞, ∞)
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Q. The function f(x) = 2x + 3 is continuous at which of the following? (2023)
A.
x = -1
B.
x = 0
C.
x = 1
D.
All of the above
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Solution
f(x) = 2x + 3 is a linear function and is continuous for all x.
Correct Answer:
D
— All of the above
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Q. The function f(x) = x^2 + 3 is continuous for which of the following intervals? (2023)
A.
(-∞, ∞)
B.
(0, 1)
C.
(1, 2)
D.
(2, 3)
Show solution
Solution
f(x) = x^2 + 3 is a polynomial function and is continuous for all x in (-∞, ∞).
Correct Answer:
A
— (-∞, ∞)
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Q. The function f(x) = x^2 is continuous at which of the following points? (2023)
A.
x = -1
B.
x = 0
C.
x = 1
D.
All of the above
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Solution
The function f(x) = x^2 is a polynomial function and is continuous at all points, including -1, 0, and 1.
Correct Answer:
D
— All of the above
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Q. The function f(x) = x^3 - 3x is continuous at which of the following points? (2023)
A.
x = -2
B.
x = 0
C.
x = 2
D.
All of the above
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Solution
The function f(x) = x^3 - 3x is a polynomial function and is continuous at all points, including -2, 0, and 2.
Correct Answer:
D
— All of the above
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Q. The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
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Solution
Left limit as x approaches 1 is 2, right limit is 1, but f(1) = 2. Hence, it is discontinuous at x = 1.
Correct Answer:
B
— No
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Q. The function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 } is:
A.
Continuous
B.
Not continuous
C.
Continuous from the left
D.
Continuous from the right
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Solution
The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.
Correct Answer:
B
— Not continuous
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Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.
Correct Answer:
B
— No
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Q. What can be said about the function f(x) = |x| at x = 0?
A.
Continuous
B.
Discontinuous
C.
Only left continuous
D.
Only right continuous
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Solution
The function f(x) = |x| is continuous at x = 0 since both left and right limits equal f(0) = 0.
Correct Answer:
A
— Continuous
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Q. What is the continuity of the function f(x) = sqrt(x) at x = 0? (2022)
A.
Continuous
B.
Not continuous
C.
Only left continuous
D.
Only right continuous
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Solution
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Correct Answer:
A
— Continuous
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Q. Which of the following functions is continuous at all points?
A.
f(x) = 1/x
B.
f(x) = x^3
C.
f(x) = sqrt(x)
D.
f(x) = tan(x)
Show solution
Solution
f(x) = x^3 is a polynomial function, which is continuous everywhere.
Correct Answer:
B
— f(x) = x^3
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Q. Which of the following functions is continuous at x = 0?
A.
f(x) = 1/x
B.
f(x) = e^x
C.
f(x) = tan(x)
D.
f(x) = 1/(x^2 + 1)
Show solution
Solution
The function f(x) = e^x is continuous everywhere, including at x = 0.
Correct Answer:
B
— f(x) = e^x
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Q. Which of the following functions is continuous on the interval [0, 1]?
A.
f(x) = 1/x
B.
f(x) = x^3
C.
f(x) = sqrt(x)
D.
f(x) = 1/(x-1)
Show solution
Solution
f(x) = x^3 is a polynomial function and is continuous on the interval [0, 1].
Correct Answer:
B
— f(x) = x^3
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Q. Which of the following statements is true about the function f(x) = 1/(x-1)? (2022)
A.
Continuous at x = 1
B.
Continuous everywhere
C.
Not continuous at x = 1
D.
Continuous at x = 0
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Solution
The function f(x) = 1/(x-1) is not continuous at x = 1 because it is undefined there.
Correct Answer:
C
— Not continuous at x = 1
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Q. Which of the following statements is true about the function f(x) = 1/(x-3)?
A.
Continuous at x = 3
B.
Continuous everywhere
C.
Not continuous at x = 3
D.
Continuous at x = 0
Show solution
Solution
The function f(x) = 1/(x-3) is not defined at x = 3, hence it is not continuous at that point.
Correct Answer:
C
— Not continuous at x = 3
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Q. Which of the following statements is true about the function f(x) = |x|?
A.
Continuous everywhere
B.
Discontinuous at x = 0
C.
Continuous only at x = 1
D.
Discontinuous everywhere
Show solution
Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer:
A
— Continuous everywhere
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Q. Which of the following statements is true regarding the function f(x) = 1/(x-3)?
A.
Continuous at x = 3
B.
Discontinuous at x = 3
C.
Continuous everywhere
D.
Discontinuous everywhere
Show solution
Solution
The function f(x) = 1/(x-3) is discontinuous at x = 3 because it is undefined at that point.
Correct Answer:
B
— Discontinuous at x = 3
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Q. Which of the following statements is true regarding the function f(x) = |x|?
A.
Continuous everywhere
B.
Discontinuous at x = 0
C.
Continuous only for x > 0
D.
Discontinuous for x < 0
Show solution
Solution
The function f(x) = |x| is continuous everywhere, including at x = 0.
Correct Answer:
A
— Continuous everywhere
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Showing 31 to 54 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!
