Re: [Gnuplot-info] Surface plotting discontinuities with splot

[Hans-Bernhard B. <HBB...@t-...>]

Am 12.04.2021 um 01:12 schrieb Helmut Haberzettl:

> I've googled around, but all the info I could find on how to plot 
> discontinuities seem to assume that the function F(x,y) is explicitly 
> given.

That would be because the entire concept of "continuity" is effectively 
meaningless for discrete data.

Let's recall the definition of the term "continuity": for any epsilon > 
0 there is a delta > 0 such that for all x with ||x - x0|| < delta, 
||f(x) - f(x0)|| < epsilon with x and x0 in the definition set of `f. 
In simpler words: no jumps in the interior (maximal open subset) of the 
definition set of f.

On discrete data sets you can always select delta smaller than the 
smallest distance of any two points in the data set, and thus force x=x0 
as the only choice.  I.e. since the definition set consists of isolated 
positions only, there is no interior for the definition to attach to.

Discrete data are thus, by definition, _always_ continuous.  That 
renders the concept tautological on them, and robs it of any meaning.