Is it reasonable to say the domain of the normal distribution is equal to its support?

Question

domain and support are not the same thing.

Per mathworld

The term domain is most commonly used to describe the set of values for which a function (map, transformation, etc.) is defined. For example, a function f(x) that is defined for real values $x \in R$ has domain $\mathbb R$, and is sometimes said to be "a function over the reals."

Per wiki

the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

Consider the normal distribution, it seems that the domain is equal to the support, is it reasonable to say this? In another word, if I ask some mathematicians if the domain of the normal distribution is equal to its support, what would they say? Yes? or, They are not comparable?

Answer 1

Technically, I believe the support is the smallest closed subset of the domain containing all points that are not mapped to zero.

But yes, both the domain and the support for the normal distribution are all real numbers.

Answer 2

The normal distribution is a distribution, not a function, so neither of the definitions you have given apply to it.

What is a distribution? There are several subtly different ways they are defined rigorously, not all of which are well-represented on wikipedia. But none of them coincides with the notion of a function. In its simplest form, a distribution is a mathematical object that assigns to each interval a number, satisfying some natural conditions. In this situation, the support is the set of points with the property that every non-empty open interval containing the point is assigned a strictly positive number.

If the distribution happens to have a density (which is a function) then the support of the distribution is essentially the same thing as the set of points on which the density function does not vanish.

Note, I purposely did not mention "domain" above since I do not know of a precise definition that applies to distributions. Maybe you can specify one if you have it in mind?