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Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.?
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Suppose f is a continuous real valued function defined on a compact me...
Proof of Existence of Element with Maximum Value
To prove that there exists an element p in X such that f(p) = M, we can consider the set A = {f(x) : x ∈ X}. Since f is continuous and X is a compact metric space, f(X) is also a compact set.
Compactness of X
The compactness of the domain X is necessary for this proof because if X were not compact, the function f might not achieve its supremum.
Existence of Maximum Value
Since A = {f(x) : x ∈ X} is a non-empty compact subset of real numbers, it must have a maximum element, denoted by M. By definition, M = sup {f(x) : x ∈ X}.
Existence of p such that f(p) = M
Since M is the supremum of f(x) for all x in X, there exists a sequence {x_n} in X such that f(x_n) → M as n → ∞. Since X is compact, there exists a convergent subsequence {x_nk} that converges to some point p in X.
Since f is continuous, f(x_nk) → f(p) as k → ∞. But we already know that f(x_nk) → M. Therefore, f(p) = M, which proves the existence of an element p in X such that f(p) = M.
To prove that there exists an element p in X such that f(p) = M, we can consider the set A = {f(x) : x ∈ X}. Since f is continuous and X is a compact metric space, f(X) is also a compact set.
Compactness of X
The compactness of the domain X is necessary for this proof because if X were not compact, the function f might not achieve its supremum.
Existence of Maximum Value
Since A = {f(x) : x ∈ X} is a non-empty compact subset of real numbers, it must have a maximum element, denoted by M. By definition, M = sup {f(x) : x ∈ X}.
Existence of p such that f(p) = M
Since M is the supremum of f(x) for all x in X, there exists a sequence {x_n} in X such that f(x_n) → M as n → ∞. Since X is compact, there exists a convergent subsequence {x_nk} that converges to some point p in X.
Since f is continuous, f(x_nk) → f(p) as k → ∞. But we already know that f(x_nk) → M. Therefore, f(p) = M, which proves the existence of an element p in X such that f(p) = M.
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Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.?.
Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Suppose f is a continuous real valued function defined on a compact metric space X and M=sup {f(x)/x€X}.then prove that there exists an element p€X such that f(p)=M.is compactness of the domain of f necessary? justify your answer.?.
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