MeshFunctions

is an option for plotting functions that specifies functions to use to determine the placement of mesh divisions.

Details

In Plot3D, the default setting MeshFunctions->{#1&,#2&} specifies that meshes corresponding to x and y coordinates should be constructed.
With the setting MeshFunctions->{m₁,m₂,…}, each function m_i defines a family of mesh divisions.
By default, the mesh divisions are taken to lie at positions giving equally spaced values of m_i[…].
The arguments supplied to the m_i and the default MeshFunctions settings are as follows:

Plot and ListLinePlot	x, y	{#1&}
ParametricPlot	x, y, u or x, y, u, v	{#3&} or {#3&,#4&}
PolarPlot and ListPolarPlot
	x, y, θ, r	(#3&)
RegionPlot	x, y	{#1&,#2&}
ContourPlot and ListContourPlot
	x, y, f	{}
DensityPlot and ListDensityPlot
	x, y, f	{#1&,#2&}
ContourPlot3D and ListContourPlot3D
	x, y, z, f	{#1&,#2&,#3&}
Plot3D and ListPlot3D	x, y, z	{#1&,#2&}
ListSurfacePlot3D	x, y, z	{#1&,#2&,#3&}
ParametricPlot3D	x, y, z, u or x, y, z, u, v	{#4&} or {#4&,#5&}
RegionPlot3D	x, y, z	{#1&,#2&,#3&}

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Put 5 mesh lines in the direction:

Show curves of constant real and imaginary parts of a function:

Show intersection points:

Define two polynomials:

Use MeshFunctions to find the intercepts:

Use MeshFunctions to find the intersections between two functions:

A case where Fubini's theorem does not hold [more info]:

Real and imaginary parts as mesh functions:

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