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https://math.stackexchange.com/questions/445735/compactly-supported-continuous-function-is-uniformly-continuous
$\begingroup$ Actually this can be answered easily if it is true that a sequence of uniformly continuous functions that converge uniformly to a continuous function, then that function necessarily uniformly continuous. $\endgroup$ – Chris Cave Feb 10 '15 at 11:52
https://www.uio.no/studier/emner/matnat/math/MAT2400/v11/ContFunc.pdf
continuity is the same. This means that a continuous function defined on a closed and bounded subset of Rn is always uniformly continuous. Proposition 2.1.2 Assume that X and Y are metric spaces. If X is com-pact, all continuous functions f : X → Y are uniformly continuous. Proof: We argue contrapositively: Assume that f is not uniformly continu-
http://www.msc.uky.edu/ken/ma570/lectures/lecture2/html/compact.htm
Definition 4: A function of topological spaces is continuous if for every open subset of , is an open subset of X. Definition 5: An open rectangle is defined to be a subset of of the form , i.e. There are real numbers and where for all i and In the case of ,...
https://en.wikipedia.org/wiki/Uniform_continuity
The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval.
https://ncatlab.org/nlab/show/compact+support
sequentially compact metric spaces are equivalently compact metric spaces. compact spaces equivalently have converging subnet of every net. sequentially compact metric spaces are totally bounded. continuous metric space valued function on compact metric space is uniformly continuous. paracompact Hausdorff spaces are normal
https://www.math.wisc.edu/~robbin/521dir/cont.pdf
It is obvious that a uniformly continuous function is continuous: if we can nd a which works for all x 0, we can nd one (the same one) which works for any particular x 0. We will see below that there are continuous functions which are not uniformly continuous. Example 5. Let S= R and f(x) = 3x+7. Then fis uniformly continuous on S. Proof. Choose ">0.
https://sites.math.washington.edu/~ebekyel/Math134/UniformContinuity.pdf
Continuity and Uniform Continuity Below I stands for any one of the intervals (a;b), [a;b), (a;b], [a;b], (a;1), [a;1), (1;b), (1 ;b], (1 ;1) = R. Let fbe a function de ned on an interval I. De nition of Continuity on an Interval: The function f is continuous on Iif it is continuous at every cin I. So,
https://www.researchgate.net/publication/259260858_Continuous_functions_with_compact_support
Continuous functions with compact support 107 (4) M is Hausdorff, if and only if, for every pair of distinct maximal ideals M and N of K there exist points a, b ∈ K such that a 6∈ M , b 6∈ N
https://www.math.ucdavis.edu/~hunter/m127c/hmwk6_solutions.pdf
• (a) If f, g are continuous functions with compact support, then they are bounded and uniformly continuous, since the functions are zero outside a compact (i.e. closed, bounded) interval, and a continuous function on a compact interval is bounded and uniformly continuous.
https://www.quora.com/What-is-the-difference-between-continuous-and-uniformly-continuous-for-a-function
Continuity of a function is purely a local property, whereas uniform continuity is a global property that applies over the whole space. A uniformly continuous function is continuous, but …
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