Continuity of Polynomials in the Complex Plane

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This Demonstration shows polynomials with zeros (or roots) of varying multiplicity in the complex plane. The object is to show that such a polynomial is a continuous function at a selected point .


Let be a complex-valued function, where . The function is continuous at a point if for every there is a such that for all points that satisfy the inequality , the inequality holds.

Assign a color to each point of the complex plane as a function of , namely the RGB color of three arguments , , (red, green, blue). If (with chosen by slider), use black. Otherwise, if , let ; if , let ; if , let .

The zeros of the polynomial can be set using the three locators, and their respective multiplicities can be selected. The movable white marks the point .

A black patch around means that . Subsequently, finding a such that the circle is inside the patch verifies continuity.

Use high resolution after setting all the arguments and parameters.


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




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

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

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