For a < b real numbers, let ƒ : [a; b] × [a; b] → R be such that (1) for each y € [a, b], the function x → f(x, y) is non-increasing and con- tinuous on [a, b], (2) for each x € [a, b], the function y ↔ f(x, y) is non-decreasing and con- tinuous on [a, b]. Prove that g(x) := f(x, x) is continuous on [a, b].
For a < b real numbers, let ƒ : [a; b] × [a; b] → R be such that (1) for each y € [a, b], the function x → f(x, y) is non-increasing and con- tinuous on [a, b], (2) for each x € [a, b], the function y ↔ f(x, y) is non-decreasing and con- tinuous on [a, b]. Prove that g(x) := f(x, x) is continuous on [a, b].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Continuous functions
![For a < b real numbers, let ƒ : [a; b] × [a; b] → R be such that
(1) for each y € [a, b], the function x → f(x, y) is non-increasing and con-
tinuous on [a, b],
(2) for each x € [a, b], the function y → f(x, y) is non-decreasing and con-
tinuous on [a, b].
Prove that g(x):= f(x,x) is continuous on [a, b].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F04645e62-540a-4456-8792-6d1e8d18d920%2Fo1qkigm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For a < b real numbers, let ƒ : [a; b] × [a; b] → R be such that
(1) for each y € [a, b], the function x → f(x, y) is non-increasing and con-
tinuous on [a, b],
(2) for each x € [a, b], the function y → f(x, y) is non-decreasing and con-
tinuous on [a, b].
Prove that g(x):= f(x,x) is continuous on [a, b].
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