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Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?
(a) f has exactly one discontinuity
(c) f has infinitely many discontinuities
(b) f has exactly two discontinuity
(d) f is continuous?
(a) f has exactly one discontinuity
(c) f has infinitely many discontinuities
(b) f has exactly two discontinuity
(d) f is continuous?
Most Upvoted Answer
Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing functio...
Understanding the Function f and its Properties
To analyze the function f defined on the set S = [0, 1) ∪ [2, 3], we must consider the implications of its strict monotonicity and the property of connectedness of its image f(S).
Strictly Increasing Function
- A strictly increasing function means that for any x1 < x2="" in="" s,="" f(x1)="" />< />
- This property ensures that f does not decrease or remain constant between any two points in its domain.
Connectedness of f(S)
- The image f(S) being connected implies that there are no 'gaps' in the range of f.
- Since S consists of two separate intervals, [0, 1) and [2, 3], f must map these intervals continuously into a connected interval in R.
Discontinuities of f
- f can have discontinuities at the point where S is not connected, specifically at the point 1 (end of [0, 1)) and the start of [2, 3].
- This suggests that there must be at least one discontinuity at x = 1.
Possible Cases of Discontinuities
- Since f is strictly increasing, it cannot have more than one discontinuity, as having two would create a disconnected image.
- Thus, the only possible discontinuity is at x = 1.
Conclusion
Based on the analysis:
- The correct statement is (a): f has exactly one discontinuity.
This conclusion follows from the behavior of strictly increasing functions and the requirement of connectedness of their image.
To analyze the function f defined on the set S = [0, 1) ∪ [2, 3], we must consider the implications of its strict monotonicity and the property of connectedness of its image f(S).
Strictly Increasing Function
- A strictly increasing function means that for any x1 < x2="" in="" s,="" f(x1)="" />< />
- This property ensures that f does not decrease or remain constant between any two points in its domain.
Connectedness of f(S)
- The image f(S) being connected implies that there are no 'gaps' in the range of f.
- Since S consists of two separate intervals, [0, 1) and [2, 3], f must map these intervals continuously into a connected interval in R.
Discontinuities of f
- f can have discontinuities at the point where S is not connected, specifically at the point 1 (end of [0, 1)) and the start of [2, 3].
- This suggests that there must be at least one discontinuity at x = 1.
Possible Cases of Discontinuities
- Since f is strictly increasing, it cannot have more than one discontinuity, as having two would create a disconnected image.
- Thus, the only possible discontinuity is at x = 1.
Conclusion
Based on the analysis:
- The correct statement is (a): f has exactly one discontinuity.
This conclusion follows from the behavior of strictly increasing functions and the requirement of connectedness of their image.
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Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous?
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Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous?.
Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous?.
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Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous?, a detailed solution for Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous? has been provided alongside types of Let S= [0, 1) cup[2,3] and f / S -> R be a strictly increasing function such that f(S) is connected. Which of the following statement is TRUE?(a) f has exactly one discontinuity(c) f has infinitely many discontinuities(b) f has exactly two discontinuity(d) f is continuous? theory, EduRev gives you an
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