When people says that function $f$ is a **particular case** of function $g$, does this mean that $f$ is a special case of $g$?

For example, people might say: "$f(x)=x^2$ is a particular case of differentiable functions".

Are "special case" and "particular case" the same thing?

From my experience, when I read about the aforementioned sentences, I can always deduce by myself that $f$ is indeed a special case of $g$. So I guess that the answer is "yes".

## Best Answer

(Turning a comment into an answer as suggested.)

To some degree, you can take them to be the same on the whole, both meaning "one of the cases". However to me they have slightly different intentions: I take "particular case" to be one case, but not necessarily special, but rather it is "convenient". And "special case" to mean a situation where we will pay a lot of attention to.

For example: "The function $f(x) = x$ is not the zero function, in particular $f(1)=1$." Here there is nothing special about the choice of $x=1$, but just one of many, perhaps convenient, to demonstrate something (in this case that $f$ is not the zero function). Another example. "Infinitely differentiable functions are nice. Let us look at the special case of polynomials..." This suggests that I want to have a focused discussion about polynomials. Or, "Finite groups can be described by a set of generators. In the special case of abelian groups, the generators can be described nicely by the fundamental theorem of finitely generated abelian groups".