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What is the mathematical expression for the definition of continuity?
- a)limx→cf(x) = f(c) ∀ c ∈ a
- b)limx→cf(x) = f(c) ∀ c ∈ (a,b)
- c)limx→cf(x) = f(c) ∀ c ∈ b
- d)limx→af(x) = f(c) ∀ c ∈ (a,b)
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
What is the mathematical expression for the definition of continuity?a...
function f defined on (a,b) is said to be continuous on (a,b) if it is continuous at every point of (a,b) i.e., if limx→cf(x) = f(c) ∀ c ∈ (a,b).
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What is the mathematical expression for the definition of continuity?a...
Understanding Continuity
Continuity of a function at a point is a fundamental concept in calculus. It ensures that the function behaves predictably without any sudden jumps or breaks.
Definition of Continuity
- A function f(x) is said to be continuous at a point c if the following condition holds:
- lim (x → c) f(x) = f(c)
Domain of Continuity
- The key aspect is that this condition must be satisfied for all points c within a certain interval.
- Option B states:
- lim (x → c) f(x) = f(c) ∀ c ∈ (a, b)
- This means that the function is continuous for every point c in the open interval (a, b), which is the correct definition of continuity.
Why Option B is Correct
- Open Interval (a, b): This indicates that c can take any value between a and b, excluding the endpoints.
- All Points c: The statement asserts that the function must be continuous at every point within this interval, which is crucial for ensuring that the function behaves smoothly across that entire range.
Why Other Options Are Incorrect
- Option A: Limits at c do not specify an interval; hence, it lacks clarity on where continuity is applied.
- Option C: Similar to A, it does not provide a specific interval for continuity.
- Option D: This option incorrectly applies the limit at the boundary (a) rather than within the interval (a, b).
Conclusion
- Therefore, option B accurately captures the essence of continuity by defining it over an interval where the function is expected to be smooth and unbroken.
Continuity of a function at a point is a fundamental concept in calculus. It ensures that the function behaves predictably without any sudden jumps or breaks.
Definition of Continuity
- A function f(x) is said to be continuous at a point c if the following condition holds:
- lim (x → c) f(x) = f(c)
Domain of Continuity
- The key aspect is that this condition must be satisfied for all points c within a certain interval.
- Option B states:
- lim (x → c) f(x) = f(c) ∀ c ∈ (a, b)
- This means that the function is continuous for every point c in the open interval (a, b), which is the correct definition of continuity.
Why Option B is Correct
- Open Interval (a, b): This indicates that c can take any value between a and b, excluding the endpoints.
- All Points c: The statement asserts that the function must be continuous at every point within this interval, which is crucial for ensuring that the function behaves smoothly across that entire range.
Why Other Options Are Incorrect
- Option A: Limits at c do not specify an interval; hence, it lacks clarity on where continuity is applied.
- Option C: Similar to A, it does not provide a specific interval for continuity.
- Option D: This option incorrectly applies the limit at the boundary (a) rather than within the interval (a, b).
Conclusion
- Therefore, option B accurately captures the essence of continuity by defining it over an interval where the function is expected to be smooth and unbroken.
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What is the mathematical expression for the definition of continuity?a)limx→cf(x) = f(c) ∀ c ∈ ab)limx→cf(x) = f(c) ∀ c ∈ (a,b)c)limx→cf(x) = f(c) ∀ c ∈ bd)limx→af(x) = f(c) ∀ c ∈ (a,b)Correct answer is option 'B'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about What is the mathematical expression for the definition of continuity?a)limx→cf(x) = f(c) ∀ c ∈ ab)limx→cf(x) = f(c) ∀ c ∈ (a,b)c)limx→cf(x) = f(c) ∀ c ∈ bd)limx→af(x) = f(c) ∀ c ∈ (a,b)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the mathematical expression for the definition of continuity?a)limx→cf(x) = f(c) ∀ c ∈ ab)limx→cf(x) = f(c) ∀ c ∈ (a,b)c)limx→cf(x) = f(c) ∀ c ∈ bd)limx→af(x) = f(c) ∀ c ∈ (a,b)Correct answer is option 'B'. Can you explain this answer?.
What is the mathematical expression for the definition of continuity?a)limx→cf(x) = f(c) ∀ c ∈ ab)limx→cf(x) = f(c) ∀ c ∈ (a,b)c)limx→cf(x) = f(c) ∀ c ∈ bd)limx→af(x) = f(c) ∀ c ∈ (a,b)Correct answer is option 'B'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about What is the mathematical expression for the definition of continuity?a)limx→cf(x) = f(c) ∀ c ∈ ab)limx→cf(x) = f(c) ∀ c ∈ (a,b)c)limx→cf(x) = f(c) ∀ c ∈ bd)limx→af(x) = f(c) ∀ c ∈ (a,b)Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the mathematical expression for the definition of continuity?a)limx→cf(x) = f(c) ∀ c ∈ ab)limx→cf(x) = f(c) ∀ c ∈ (a,b)c)limx→cf(x) = f(c) ∀ c ∈ bd)limx→af(x) = f(c) ∀ c ∈ (a,b)Correct answer is option 'B'. Can you explain this answer?.
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