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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.

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Formally there is no difference. In fact any family of objects is a function from index set into collection of this object. But we use both notations because sometimes we want to emphasize different points of view. For example, we often think of sequences in $\mathbb{R}$, but of course we can consider them as function from $\mathbb{N}$ into $\mathbb{R}$. – Norbert Aug 3 '12 at 10:56
To Norbert's comment one might add the following: In a family $(U_\alpha)_{\alpha\in I}$ the index set $I$ and the implied function $f:\, I\to ?$ serves only to structure the list of objects $U_\alpha$; it is not intended as an operation that moves points from one space to another. – Christian Blatter Aug 3 '12 at 14:34
up vote 1 down vote accepted

Let $S$ be a set of mathematical objects that you would like to parameterize. Then a family of objects in $S$, parameterized by a set $T$, is a function $f : T \to S$. Often we want to require additional properties of this function, e.g. continuity or smoothness with respect to suitable topologies on $T$ and $S$. For one-parameter families, $T = \mathbb{R}$.

The difference between this and a function is a matter of emphasis: the emphasis is on $S$ and not on $T$. More precisely, you want to think of $f$ as a generalized point of $S$ in the sense of the Yoneda lemma (you get an ordinary point if $T$ is itself a point).

Sometimes it is nontrivial to write down the set $S$ with appropriate structure. For example, roughly speaking a vector bundle over a space $X$ is an $X$-parameterized family of vector spaces, but we don't (always) define such a thing by defining a space of vector spaces and mapping $X$ into it (although we can!). Instead a vector bundle is generally defined as a space $B$ with a suitable type of continuous map $f : B \to X$ such that the fibers $f^{-1}(x), x \in X$ have the structure of vector spaces (and satisfying some other conditions). This is also the strategy taken in algebraic geometry, e.g. to define a family of elliptic curves.

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