Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
This is regarding continuity of a function?
What are examples of functions of one [ also 2] variable[s] which are continuous but not uniformly continuous. I shall appriciate if an answerer gives the value of x in the domain, clearly. Secondly in which text book will I get the proof of the theorem that every vontinuous function on [a, b] attains its bounds in [a b]. Many thanks in advance.
1 Answer
- ?Lv 61 decade agoFavorite Answer
A. 1. f(x) = x^2 is not uniformly continuous on [0, oo). The basic reason is that
its derivative is positive and unbounded. Consider any fixed h > 0.
|f(x+h) - f(x)| = 2xh + h^2 > 1 if x > 1/(2h). Therefore, with epsilon = 1, no h
works over the entire infinite interval.
2. g(x) =1/x is not uniformly continuous on (0,1].
3. h(x) = cos(pi/x) is not uniformly continuous on (0,1]. h(1/n) = (-1)^n
so |h(1/n) - h(1/(n+1)| = 2, while 1/n - 1/(n+1) ---> 0 as n --> oo
--------
B. Analogues of examples above: F(x,y) = x^2 + y^2.
G(x,y) = 1/(x^2 + y^2), domain all (x,y) except (0,0).
--------------
C. Any textbook called "Introduction to Analysis" will have this proof.
I like the book by Ken Ross, whose title might be slightly different.
