Exposition Style
A one-paragraph note on how to write a piecewise function so that nobody can misread it.
Crocker, S.D., "Exposition Style", RFC 154, DOI 10.17487/RFC0154, May 1971, <https://www.rfc-editor.org/info/rfc154>.
@misc{rfc154,
series = {Request for Comments},
number = 154,
howpublished = {RFC 154},
publisher = {RFC Editor},
doi = {10.17487/RFC0154},
url = {https://www.rfc-editor.org/info/rfc154},
author = {S.D. Crocker},
title = {{Exposition Style}},
pagetotal = 1,
year = 1971,
month = may,
} RFC text
Network Working Group S. Crocker
Request for Comments: #154 UCLA
NIC: #6759 12 May 1971
Categories: C.4
Updates: #107
Obsoletes: #132
Exposition Style
As a pedagogical device for describing functions such as the one
below
|
| +-----------------
| /:
| / :
| / :
| / :
| / :
| / :
-----+------------------------------------
| a
|
where two formulae, f1 and f2, are necessary for adjoining domains but
the function is continous at the boundary point, I usually write the
description in the form
f(x) = f1(x) for x =< a
f(x) = f2(x) for x >= a
The astute reader will note that the domains overlap, but that
f1(a) = f2(a), so no semantic ambiguity obtains.
[ This RFC was put into machine readable form for entry ]
[ into the online RFC archives by Naoki Matsuhira 5/97 ]
AI ELI5
On this page
§1The Problem Being Described
RFC 154 is a one-page note by S. Crocker, dated 12 May 1971, titled Exposition Style. It calls its subject a pedagogical device, a way of teaching or describing a function.
The function it has in mind is drawn as a graph: a line that runs flat and then, at a point marked a on the horizontal axis, bends upward. Two formulae, f1 and f2, are needed because each one covers a different adjoining domain, but the function is continuous at the boundary point.
Imagine a slide that is flat for a while and then goes up like a ramp. You need one rule for the flat part and another rule for the ramp.
§2The Suggested Form
The author usually writes the description with two lines:
f(x) = f1(x) for x =< a f(x) = f2(x) for x >= a
Both lines include the boundary point a, so the two domains overlap there.
You write one rule for everything up to the meeting spot and another rule for everything from the meeting spot onward.
§3Why the Overlap Is Safe
The RFC points out that an astute reader will notice the overlap. Normally two formulas claiming the same point could disagree about its value.
Here they cannot, because f1(a) = f2(a). Whichever line a reader applies at a, the answer is the same, so in the RFC's words no semantic ambiguity obtains.
If two friends both say a spot is theirs but they agree on what is there, nobody gets confused.