Exposition Style
A short note on writing two formulas for a continuous piecewise function.
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 ]
ISO Plain Language
On this page
§1A Function with Two Formulas
The note describes a function that needs two formulas, f1 and f2, on adjoining domains. The function is continuous at the boundary point a.
It presents a graph in which the function changes at a while remaining continuous there.
Some functions use one formula on one side of a boundary and a different formula on the other. This note explains how to write that definition.
§1The Suggested Notation
The suggested form is f(x) = f1(x) for x <= a, and f(x) = f2(x) for x >= a. Both conditions include the boundary point a.
The RFC observes that a careful reader will notice the domains overlap at that point.
Write the first rule for values up to and including a. Write the second rule for values from a onward.
§1Why the Overlap Is Unambiguous
Two formulas can overlap at a only if they give the same value there. The note states that f1(a) = f2(a).
Because both formulas agree at the shared boundary, the note says the overlap creates no semantic ambiguity.
At the shared point, both rules give the same answer. A reader can use either rule there without getting a different value.