Add function values in the 2D point table:

Test interpolation in one point:

Often one obtain the same value if the option Type is specified as Undirected:

Check with the original function in the same point:

If the same interpolation should be used in many different points, the function can be precompiled without specifying in which point it will be used. With InterpolationOrder set to 1, the interpolated function is continuous. Note, in the plot below, that extrapolation is used along the border of the plot.

The same plot with InterpolationOrder set to 2, with continuous derivative.

The same plot with InterpolationOrder set to 3, with continuous second order derivative.

The same plot with InterpolationOrder set to 0, with piecewise constant interpolated values. The plotting routine in

*Mathematica* is not able to make the edges straight with the default settings.

Let us check some 1D cases

Add function values in the point table:

Test interpolation with default settings:

Check interpolation with the normal

Interpolation: