Summary

DescriptionNon-holomorphic complex conjugate.svg	English: Diagram showing why the function ${\displaystyle f(z)={\bar {z}}}$ is not holomorphic. Along the real axis, it is equal to the function g(z) = z, and the limit of the slope as it approaches zero is 1. Along the imaginary axis, it is equal to the function g(z) = -z, and the limit of the slope as it approaches zero is -1. Other angles give yet different limits. Without a single matching limit, the function is not differentiable. Source below.
Date	25 March 2013
Source	Own work   This W3C-unspecified diagram was created with Mathematica.
Author	User:Dcoetzee

Licensing

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The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse

Source

Mathematica source:

Show[Plot[{x, -x}, {x, -2, 2}, PlotStyle -> Black], 
 ListPlot[{{1, 0}, {-1, 0}, {0, 1}, {0, -1},
   {1, 1}, {-1, -1}, {-1, 1}, {1, -1}},
  PlotStyle -> PointSize[Large]],
 AxesStyle -> FontSize -> 14, AspectRatio -> 1, 
 PlotRange -> {{-2, 2}, {-2, 2}}]

LaTeX source:

$$ \frac{\bar{z} - \bar{0}}{z-0} $$
$$ 1 $$
$$ i $$
$$ -1 $$
$$ -i $$

Converted to SVG using [1] and embedded in the SVG with Inkscape. Axes labels from Mathematica removed in Inkscape.