Are custom named functions acceptable notation?

Question

A custom name being, for example, my function name (MFN):

$MFN(x) := ax + b$

As contrasted with:

$\delta(x) := ax + b$

Questions:

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

Answer 1

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.