Interpolation, Method -> "Delaunay"

Some 2D points:

Add function values in the 2D point table:

Test interpolation in one point:

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. The interpolated function is continuous. According to the ordinary definition of the Delaunay interpolation, the method is not specified outside the convex hull (drawn in white) of the interpolation points. To avoid error messages during the generation of the plots, like that below, I have defined extrapolated values. By drawing the outer bisectors from the convex hull points, and interpolating linearly along lines parallel to the edges of the convex hull, continuous extrapolated values can be found.