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https://math.stackexchange.com/questions/110573/continuous-mapping-on-a-compact-metric-space-is-uniformly-continuous
Continuous mapping on a compact metric space is uniformly continuous. Ask Question Asked 7 years, ... Continuous with compact support implies uniform continuity. 2. ... Topological space of continuous function is not compact. 1.
http://www.msc.uky.edu/ken/ma570/lectures/lecture2/html/compact.htm
Lecture 2: Compact Sets and Continuous Functions 2.1 Topological Preliminaries. What does it mean for a function to be continuous? An elementary calculus course would define: Definition 1: Let and be a function. Let and . The function has limit as x approaches a if for every , there is a such that for every with , one has . This is expressed as
https://en.wikipedia.org/wiki/Uniform_continuity
More generally, a continuous function : → whose restriction to every bounded subset of S is uniformly continuous is extendable to X, and the converse holds if X is locally compact. A typical application of the extendability of a uniformly continuous function is the proof of the inverse Fourier transformation formula. We first prove that the ...
https://www.math.wisc.edu/~robbin/521dir/cont.pdf
Example 15. The function f(x) = p xis uniformly continuous on the set S= (0;1). Remark 16. This example shows that a function can be uniformly contin-uous on a set even though it does not satisfy a Lipschitz inequality on that set, i.e. the method of Theorem 8 is not the only method for proving a function uniformly continuous.
http://www.math.jhu.edu/%7Efspinu/405/405-continuity%20thms.pdf
continuous (proved in class), f(D) µ R¡f0g, hence h = g –f is continuous. 3. General properties continuous functions 3.1. Theorem. A continuous function maps compact sets into compact sets. Proof. In other words, assume f: D ! Ris continuous and D is compact. Then we need to prove that the image f(D) is a compact subset of R.
https://en.wikipedia.org/wiki/Continuous_(topology)
Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.
https://mathoverflow.net/questions/141660/continuous-functions-as-uniformly-continuous-function
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