Function of a real variable
function whose domain a subset of real numbers
In mathematics, a function of a real variable is a function whose domain is a subset of
R
{\displaystyle \mathbb {R} }
. Many real functions that are often encountered have their domain contain an interval of non-empty interior, and may be continuous, or have some degree of smoothness, over one or more intervals, each of non-empty interior, in the domain. In older texts, the theory of functions of a real variable is often synonymous with what is usually now called real analysis.
The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of
R
{\displaystyle \mathbb {R} }
-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an
R
{\displaystyle \mathbb {R} }
-algebra, such as the complex numbers or the quaternions. The structure
R
{\displaystyle \mathbb {R} }
-vector space of the codomain induces a structure of
R
{\displaystyle \mathbb {R} }
-vector space on the functions. If the codomain has a structure of
R
{\displaystyle \mathbb {R} }
-algebra, the same is true for the functions.
The image of a function of a real variable is a curve in the codomain.
“Function of a real variable” enters the record as function whose domain a subset of real numbers. Crown Archives preserves that source wording while asking what Function, real and variable can confirm, complicate or overturn.
Why this record matters
“Function of a real variable” is worth following because a concise public description often conceals a longer documentary argument. Here, Function, real and variable provides the most credible route into that argument.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Jun 6, 2026. The linked authority identifier is Q861681. The Library of Congress control number is sh85052357. 1 of 1 selected statements include explicit references; 0 carry qualifiers and 0 use preferred rank.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The source lead contains qualifying language; that uncertainty should survive quotation, summary and reuse. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Function of a real variable”, its source revision and the description used here.
- Expand the search: follow Function of a real variable primary sources, Function of a real variable archive and Function research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Function of a real variable”?
- Which cited source is closest to the event, object or claim?
- Which institution is responsible for the underlying evidence?
Search terms from this dossier
This entry incorporates text from “Function of a real variable” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.