Term for an example that doesn't satisfy the conditions of a theorem but is presented in order to illustrate necessity of those conditions? When a theorem is presented, there will often be accompanying examples, e.g.

Theorem: If $A$ and $B$, then $Q$.
Example 1: A case where $A$ and $B$ are satisfied, so $Q$ is also satisfied.
Example 2: A case where $A$ is satisfied but $B$ is not satisfied, and $Q$ is also not satisfied.

Example 1 is an example of the application of the theorem. Example 2 isn't an application of the theorem, but it illustrates the need for the conditions in the theorem. Such examples can help us understand why the theorem is necessary in the first place.
Is there a word or phrase to describe this class of example? It is not a "counterexample" because it doesn't disprove the theorem.  It is instead a counterexample to a weaker version of the theorem that's no longer correct. It's not really an "inverse" either because we aren't asserting that ${\sim}(A \text{ and } B)$ implies ${\sim}Q$. Such examples might be preceded with the phrases "however," or "on the other hand".

For a simple concrete example:

Theorem: If $p$ is prime then $\mathbb Z_p$ is a field.
For example $\mathbb Z_3$ is a field: verify that all its elements have multiplicative inverses. On the other hand, $\mathbb Z_4$ is not a field since the element $2$ does not have an inverse.

For another example, this answer prompted me to ask this question.
 A: To the best of my knowledge, no such term exists.  I would even go a step farther, and argue that the terms "example" and "counterexample" are syntactically different in mathematics, and that the word you are looking for is subsumed by the word "example".


*

*"Example" is informal, and has the same meaning in mathematics as it has in everyday English.  This is a non-technical term which is, nevertheless, quite useful.  The term is used mostly in informal exposition, and as a header to a block of text (in the same way that "Definition", "Theorem", or "Proposition" are often used to introduce a block of text).

*On the other hand, "counterexample" does actually have some technical meaning in mathematics—it is a refutation of the immediately preceding statement.  A counterexample is not actually an example, as such—rather, a counterexample is a method of disproof.
Personally, I think it best to avoid introducing new terminology unless you need to.  Thus, I would continue to use the word "example" for the kinds of examples that you are talking about.  This is particularly true for the kinds of text blocks mentioned above.  That is, I would continue to typeset things like

Example: Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nulla finibus tellus vitae nisi venenatis, sit amet condimentum erat ullamcorper. Proin at nibh tincidunt, molestie nisl eget, sagittis lacus. In lectus sem, pellentesque quis volutpat id, imperdiet id magna.

That being said, I can think of few ways of adding an adjective which might make your point:


*

*Non-Example

*Illuminating Example

*Illustrative Example

*Partial Example

*Motivating Example[1]

*Example of Sufficiency / Example of Necessity

[1] Often an example is presented before the main result.  These kinds of example often demonstrate how an obvious or naive statement of the result fails, and motivate the introduction of extra hypotheses which will allow a result to carry through.  Thus these kinds of examples are motivating.
