A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is regarded as an "identity" when one of their domains is a proper subset of the other, and the two functions coincide on the smaller domain. Examples of such equalities are abundant in high school mathematics exercises, e.g. $\frac{x^2-1}{x-1}=x+1,\ e^{\log x}=x$ etc.. Often, the domains are not specified in the exercises.
A somewhat similar but subtly different case is when both functions are defined on the same domain but whether they are equal depends on the exact domain. For instance, $e^{x+y}\equiv e^xe^y$ for real or complex numbers but not for quaternions. Yet, for the purpose of discussion, let us focus on the aforementioned case of $f$ and $g$. For pedagogical purposes:
- Do you consider $f=g$ an "identity"? What does an identity mean?
- How to convince high school students that your definition is a good one?
