(1.2) The collection of functions from R to R {f, f2,** , fn} is uniformly continuous on compacta if and only if for each compact subset K of R, and for every positive number e there exists a positive number 8 such that if x and y are elements of K which differ in absolute value by less than 8, then for each i E {1,2, …,n} the function values f:(x) and f;(y) differ in absolute value by less than e.
(1.2) The collection of functions from R to R {f, f2,** , fn} is uniformly continuous on compacta if and only if for each compact subset K of R, and for every positive number e there exists a positive number 8 such that if x and y are elements of K which differ in absolute value by less than 8, then for each i E {1,2, …,n} the function values f:(x) and f;(y) differ in absolute value by less than e.
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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please just answer (1.2) If you can answer both i would be thankful so i can compare my answers to both problems with their solutions but i am mainly posting this for the second subpart (1.2). Thank you.
![(1)
For each of the following definitions, (a) rewrite the definition using math symbols
wherever possible.; (b) write the negation of the definition using math symbols wherever
possible; (c) write the negation of the definition in English.
(1.1) A function f: A → B is open if and only if whenever U is an open subset of A, the image of U
under f, f (U), is an open subset of B.
(1.2) The collection of functions from R to R {fu f2,*, fn} is uniformly continuous on compacta if
...
and only if for each compact subset K of R, and for every positive number e there exists a
positive number 8 such that if x and y are elements of K which differ in absolute value by less
than 8, then for each i E {1,2,·,n} the function values fi (x) and f:(y) differ in absolute value
•..
by less than E.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6214fda8-f992-4e88-b320-5195339361f5%2Fc51661da-e140-4d34-b009-44802f33c8fe%2F1o93gp9_processed.png&w=3840&q=75)
Transcribed Image Text:(1)
For each of the following definitions, (a) rewrite the definition using math symbols
wherever possible.; (b) write the negation of the definition using math symbols wherever
possible; (c) write the negation of the definition in English.
(1.1) A function f: A → B is open if and only if whenever U is an open subset of A, the image of U
under f, f (U), is an open subset of B.
(1.2) The collection of functions from R to R {fu f2,*, fn} is uniformly continuous on compacta if
...
and only if for each compact subset K of R, and for every positive number e there exists a
positive number 8 such that if x and y are elements of K which differ in absolute value by less
than 8, then for each i E {1,2,·,n} the function values fi (x) and f:(y) differ in absolute value
•..
by less than E.
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