Better terminology than "equivalence class of functions" Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions.  For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for all $t \in D$} \} $$ be the equivalence class of functions which agree with $f$ on the set $D$.
I refer to this object frequently in a paper I am writing.  The notation $[f]_D$ is short and sweet, but the terminology "equivalence class of functions" is too clunky.  
Question:  Is there a better name for the object $[f]_D$?
 A: For ease of exposition, it is often good to supress those parameters that do not vary often.  In the fragment suggested by Tom LaGatta in the comments, there is enough explanatory information and samples of forms of usage that I am quite comfortable with it as a first, second, or even third paragraph.  I, too, would get tired of writing or reading 10 such paragraphs.  So I would set up a context paragraph and occasional reminder sentences, such as follows (forgive the lack of Tex):
"Since D won't change much in the following, we will write [f] sometimes to hide the dependence on D.  Also, we will use the nickname 'class' sometimes to refer to the fact that [f]_D is an equivalence class of functions."
And then occasionally
" so we have that expression(h) does this for us (remember [h] is short for the class [h]_D)..."
I hope this helps.
Gerhard "Ask Me About System Design" Paseman, 2011.08.26
A: What about "an extension class" or "an extension family" (from D)?
A: How about "$D$-class"?
A: Why don't you just say that for any function $\phi\in C(D)$ there is a measure $P_\phi$ on the class $[\phi]$ of functions $f\in C(\mathbb R)$ whose restriction $f|_D$ coincides with $\phi$?
