Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities.
This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b.
[2] One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis): Despite this fact, one could attempt to define a discontinuous function f(x) by specifying that f(x) = 1 for x = 0, and f(x) = 0 for x ≠ 0.
If the law of the excluded middle held, then this would be a fully defined, discontinuous function.
We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line.