Below you will find example sentences with "uniformly continuous". The examples show how this phrase is used in natural context and which words often surround it.
Uniformly Continuous in a sentence
Corpus data
- Displayed example sentences: 15
- Discovered as a combination around: continuous
- Corpus frequency in the collocation scan: 12
- Phrase length: 2 words
- Average sentence length: 22.4 words
Sentence profile
- Phrase position: 3 start, 2 middle, 10 end
- Sentence types: 15 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "uniformly continuous" has 2 words and usually appears near the end in these examples. The average sentence has 22.4 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as function is uniformly continuous, a uniformly continuous function is, function, uniform and interval stand out.
- In the phrase index, this combination connects with continuous functions, continuous function, continuous improvement, continuous functions, continuous function and continuous improvement, linking the page to nearby combinations.
Example types with uniformly continuous
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
Any absolutely continuous function is uniformly continuous. (7 words)
More generally, every Hölder continuous function is uniformly continuous. (9 words)
All uniformly continuous functions are continuous with respect to the induced topologies. (12 words)
The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain. (34 words)
History The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. (33 words)
That is, the discrete space X is free on the set X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps. (32 words)
Example sentences (15)
All uniformly continuous functions are continuous with respect to the induced topologies.
Any absolutely continuous function is uniformly continuous.
A uniformity compatible with the topology of a completely regular space X can be defined as the coarsest uniformity that makes all continuous real-valued functions on X uniformly continuous.
Central to the theory was the concept of uniform continuity and the theorem stating that every continuous function on a closed interval is uniformly continuous.
History The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous.
If a real-valued function is continuous on and exists (and is finite), then is uniformly continuous.
Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function.
In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval.
More generally, every Hölder continuous function is uniformly continuous.
On the other hand, the Cantor function is uniformly continuous but not absolutely continuous.
That is, the discrete space X is free on the set X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps.
The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous.
The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.
A uniformly continuous function is defined as one where inverse images of entourages are again entourages, or equivalently, one where the inverse images of uniform covers are again uniform covers.
The image of a totally bounded subset under a uniformly continuous function is totally bounded.