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https://math.stackexchange.com/questions/445735/compactly-supported-continuous-function-is-uniformly-continuous
Compactly supported continuous function is uniformly continuous. Ask Question Asked 6 years, ... Is it true that a continuous function with compact support is uniformly continuous? 0. ... $ are not. 2. How is it not the case that every continuous function is uniformly continuous? 4.
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://www.math.ucdavis.edu/~hunter/m127c/hmwk6_solutions.pdf
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://mathoverflow.net/questions/3764/does-there-exist-a-continuous-function-of-compact-support-with-fourier-transform
Continuous compactly supported functions are in L^1 and so their Fourier Transform (FT) is bounded. So everything depends on the behaviour at infty. Observe that the characteristic function of a bounded interval is not (absolutely) integrable, being "almost" sen(y)/y.
https://www.academia.edu/11892426/A_Converse_To_Continuous_On_A_Compact_Set_Implies_Uniform_Continuity
It is well known that on a compact metric space, continuous functions are uniformly continuous. However, the converse is not true. If every continuous function is uniformly continuous on a metric space, what can we say about that space?
https://ncatlab.org/nlab/show/compact+support
continuous metric space valued function on compact metric space is uniformly continuous. paracompact Hausdorff spaces are normal. ... has compact support (or is compactly supported) if the closure of its support, the set of points where it is non-zero, is a compact subset.
https://www.uio.no/studier/emner/matnat/math/MAT2400/v11/ContFunc.pdf
If the underlying space X is compact, pointwise continuity and uniform continuity is the same. This means that a continuous function defined on a closed and bounded subset of R n is always uniformly continuous.
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://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
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