This post is rather short and concerned with terminology. I hope it is still a valid question, I shall try to make it as clear and precise as possible!
What is the difference between a one - parameter family, and a function of one variable ?
To add a little bit more:
In my recent studies I frequently come upon the concept of a parameterized family of mathematical objects, for example one can describe the Schwartz kernel of a pseudodifferential operator as a parameterized family of tempered distributions. More generally I often encounter texts in which (one - parameter) families of operators are mentioned.
In most cases these families are cited without further comments as to their precise meaning, presumably because it is taken for granted that the reader is aware of it.
One thing, for example, that I guess might be different between functions and parameterized families is that the former comes with a domain and codomain, whereas in the latter case I am not quite sure whether this is part of the data.. ?
In case this question is too broad to be answered here I would also be very thankful about reference suggestions.