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When we come to consider continuity, differentiability and smoothness for
relations – notably including many-valued functions
such as square
root and, in the complex plane, logarithm – we need to take account of
there being potentially several left values related to any given right value,
rather than the unique left value a mapping relates to any given right value.
Fortunately, the usual inverse image of open is open
formulation of
continuity interacts well with this, although matters get a little trickier when
we come to derivatives.
Now, as ever, I'm trying to avoid the challenge-response protocols inherent
in the inverse image of open is ipen
specification of continuity, so I
need to think about how my alternative approaches
can be applied to relations; which, in any case, fits well with what I'm going
to have to think about in order to apply my approach to
differentiation to relations, likewise. In each case I'm looking at a simplex
in the input space and the simplices in a result space (the image space or the
space of gradients of chords) that contains all results from within the input
simplex. We can intersect all such result space simplices that contain all the
results from any input space simplex with a given input in the interior and ask
whether the intersection is a single result; if so, we pass the test.