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26 Mar 2013 Method one: Note that (a,b) is dense in [a,b], and that any uniformly continuous function on a dense subset of a space can be extended to a
4 Dec 2011 The theorem you mention is kind of strange. You don't need to assume uniform continuity, it is enough to suppose that your function f is
Bulletin 53 (2004), 53–56. 53. When Uniformly-continuous Implies Bounded. ANTHONY G. O'FARRELL1. 1. Introduction. Let (X, ?) and (Y,?) be metric spaces.
15 Apr 2013 Because function is periodic with period T?0, we can consider function on [0,T]. If it not bounded on R, it must not bounded on [0,T], but
24 Jun 2013 Hint: Any uniformly continuous function on a dense set can be extended continuously on the whole set.
17 Jun 2017 If B?R is bounded, then ?B is compact. Since f is uniformly continuous, there is a unique continuous extension ?f of f defined on ?B. Since ?B is
8 Dec 2015 If a function f:[0,1)>R is uniformly continuous, then for any ?>0, there is a ? such that if x,y?[0,1) are ? apart, they will have images in R that will
F(x)=x is unbounded and uniformly continuous. I think any unbounded, continuous function with a bounded derivative will be uniformly
4 Dec 2011 f is presumably a real-valued function (the value of f at any single point is finite); so, the f(zi) are fixed and, in particular, bounded numbers
15 Nov 2013 Hint: Since f is uniformly continuous, there is a constant M so that Since A is bounded, you can cover it with a finite number of such intervals.
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