Continuity of a Complex Function

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Let be a complex function where and are open subsets in . The function is continuous at the point if for every there is a such that for all points that satisfy the inequality , the inequality holds.

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We assign a color to each point of the complex plane as a function of , namely the RGB color of four arguments , , , and (red, green, blue, and opacity). If (with chosen by the slider), we use black. Otherwise, if , ; if , ; if , .

A black patch around the point means that the function has . After that, we find a such that the circle is inside the patch. Note the branch cut for along the negative real axis.

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Contributed by: Izidor Hafner (February 2016)
Open content licensed under CC BY-NC-SA


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References

[1] F. A. Farris, Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, Princeton: Princeton University Press, 2015 pp. 35–36.

[2] A. G. Sveshnikov and A. N. Tikhonov, The Theory of Functions of a Complex Variable, Moscow: Mir Publishers, 1971 pp. 24–25.



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