What is the word for an arbitrary simple example, typically used with proofs?
Typical usage is with math, or philosophy, proofs. Also typically the simple example disproves the theory, but is of a arbitrarily contrived nature and not something that would naturally arise.
Is that clear enough?
A mathy example would be, in differential equations. If Brent said all functions are differentiable. The Professor X would say what about f(x)=x^2 for all x not equal to 3, when f(3) = -100. Then f(3) is a jump discontinuity and is not differentiable. This is an arbitrary, simple example disproving Brent's statement.
The example is ______
Solution 1:
The phrase you're thinking of is pathological example.
This is used in maths for an unnaturally contrived situation, often devised just to disprove some theory which usually does hold true.
For instance, pretty much any function that's continuous everywhere is also differentiable somewhere (and indeed almost everywhere). But that's not true as a theorem, without the "pretty much", because of functions like the blancmange function or Weierstrass's function which are continuous everywhere and differentiable nowhere.
Source: I'm a mathematician :-) Also, Wikipedia link.
Solution 2:
I think you're talking about multiple possible things here.
I've heard the word 'toy example' or 'toy problem' (see Wikipedia and Google) to denote simple (or simplified) problems demonstrating the application of a theory or technique in the natural sciences.
Since proofs are supposed to cover all cases, not just simple ones, I'm not sure they're so relevant to proofs. I suppose in the very specific case of proofs by mathematical induction (https://en.wikipedia.org/wiki/Mathematical_induction), the 'base case' is something resembling what you originally describe.
A counterexample that decisively disproves something may be a refutation (e.g., 'That example serves as refutation of the theory.')
Simple examples are relevant to arguments, as opposed to merely proofs, are often referred to as 'thought experiments' in philosophy. See https://en.wikipedia.org/wiki/Trolley_problem for a popular example. Or, there's always the ever popular Schrodinger's cat.