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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?
15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x
(a) f has exactly one discontinuity
(c) f has infinitely many discontinuities
(b) f has exactly two discontinuity
(d) f is continuous?
15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x
(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 on Set S
The set S = [0, 1) ∪ [2, 3] consists of two disjoint intervals. A strictly increasing function f defined on S suggests that as x increases, f(x) also increases.
Properties of Connectedness
- The image f(S) must be connected.
- A strictly increasing function can only map disjoint intervals to non-overlapping ranges.
- Hence, f([0, 1)) and f([2, 3]) must be separate but maintain a connected output.
Discontinuities of f
Given that f is strictly increasing, we analyze the possible discontinuities:
- At x = 1: The function transitions from the first interval [0, 1) to the second interval [2, 3].
- Since f is strictly increasing, f(1) must be less than f(2). Thus, there is a jump discontinuity at x = 1.
Analysis of Options
- Option (a): f has exactly one discontinuity. This is true since the only discontinuity arises at the transition point x = 1.
- Option (b): f has exactly two discontinuities. This is false.
- Option (c): f has infinitely many discontinuities. This is also false.
- Option (d): f is continuous. This is false, as we established a jump discontinuity.
Conclusion
Thus, the correct answer is:
- (a) f has exactly one discontinuity.
This conclusion stems from the nature of strict monotonicity and the structure of the set S. The only point of discontinuity occurs at the boundary between the two intervals.
The set S = [0, 1) ∪ [2, 3] consists of two disjoint intervals. A strictly increasing function f defined on S suggests that as x increases, f(x) also increases.
Properties of Connectedness
- The image f(S) must be connected.
- A strictly increasing function can only map disjoint intervals to non-overlapping ranges.
- Hence, f([0, 1)) and f([2, 3]) must be separate but maintain a connected output.
Discontinuities of f
Given that f is strictly increasing, we analyze the possible discontinuities:
- At x = 1: The function transitions from the first interval [0, 1) to the second interval [2, 3].
- Since f is strictly increasing, f(1) must be less than f(2). Thus, there is a jump discontinuity at x = 1.
Analysis of Options
- Option (a): f has exactly one discontinuity. This is true since the only discontinuity arises at the transition point x = 1.
- Option (b): f has exactly two discontinuities. This is false.
- Option (c): f has infinitely many discontinuities. This is also false.
- Option (d): f is continuous. This is false, as we established a jump discontinuity.
Conclusion
Thus, the correct answer is:
- (a) f has exactly one discontinuity.
This conclusion stems from the nature of strict monotonicity and the structure of the set S. The only point of discontinuity occurs at the boundary between the two intervals.
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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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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?15 Let where x \in (0, 1) Then on (0.1) f' * (x) = (e ^ (- 1/x))/x(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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