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https://www.planetmath.org/SmoothFunctionsWithCompactSupport
Then the set of smooth functions with compact support (in U) is the set of functions f: ℝ n → ℂ which are smooth (i.e., ∂ α f: ℝ n → ℂ is a continuous function for all multi-indices α) and supp f is compact and contained in U. This function space is denoted by C 0 ∞ (U).
https://math.stackexchange.com/questions/67370/smooth-functions-with-compact-support-are-dense-in-l1
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https://math.rice.edu/~semmes/fun5.pdf
Some basic aspects of smooth functions and distributions on open subsets of Rn are briefly discussed. Contents 1 Smooth functions 2 2 Supremum seminorms 3 3 Countably many seminorms 4 4 Cauchy sequences 5 5 Compact support 6 6 Inductive limits 8 7 Distributions 9 8 Differentiation of distributions 10 9 Multiplication by smooth functions 11
https://www.encyclopediaofmath.org/index.php/Function_of_compact_support
The support of is the closure of the set of points for which is different from zero . Thus one can also say that a function of compact support in is a function defined on such that its support is a closed bounded set located at a distance from the boundary of by a number greater than , …
https://mathoverflow.net/questions/237636/are-compactly-supported-continuous-functions-dense-in-the-continuous-functions-o
Continuous functions on $\mathbb R^d$ such that the support is a compact subset of $\overline{\Omega}$? For "nice" $\Omega$ this would be the space of continuous functions on $\Omega$ vanishing at the boundary. $\endgroup$ – Jochen Wengenroth Apr 29 '16 at 12:50
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