TITLE: Are custom named functions acceptable notation?
QUESTION [5 upvotes]: A custom name being, for example, my function name (MFN):
$MFN(x) := ax + b$
As contrasted with:
$\delta(x) := ax + b$
Questions:

Is it permissible to name the function $MFN$ above? Or is this restricted to very well known functions, such as $sgn(x)$?
Can you refer me to a source for the use of word abbreviations as names of functions?
Is lower-case preferred to upper-case? What circumstances dictate upper-case function names (or letters)?

REPLY [2 votes]: The knee-jerk answer is that you can use any notation as long as you define it before you use it.
But mathematicians generally prefer single-letter function names, which is part of the reason why certain letters of the Greek alphabet have more than one function attached to them (e.g., Carmichael's and Liouville's $\lambda(n)$).
I thought I had seen an example of a function name with two capital letters on Mathworld but now I can't find it. The problem with using two or three lowercase letters is that it might look like a standard function, like $\log x$ or $\sin y$.
From what I've seen over the years, the preference for custom function symbols goes something like this:

a single lowercase Latin letter
a single Greek letter
two uppercase Latin letters
three uppercase Latin letters
two or three lowercase Latin letters

If you choose to use more than one Latin letter, it's best not to italicize them as it looks like a title.