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

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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.