Below you will find example sentences with "continuous functions". The examples show how this phrase is used in natural context and which words often surround it.
Continuous Functions in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: continuous
- Corpus frequency in the collocation scan: 12
- Phrase length: 2 words
- Average sentence length: 22.5 words
Sentence profile
- Phrase position: 6 start, 8 middle, 6 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "continuous functions" has 2 words and usually appears in the middle in these examples. The average sentence has 22.5 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as all uniformly continuous functions are continuous, continuous functions and homeomorphisms, space, uniformly and transforms stand out.
- In the phrase index, this combination connects with recursive functions, hash functions, uniformly continuous, uniformly continuous, continuous function and continuous improvement, linking the page to nearby combinations.
Example types with continuous functions
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. (9 words)
Antiderivatives of non-continuous functions Non-continuous functions can have antiderivatives. (11 words)
Continuous functions are Borel functions but not all Borel functions are continuous. (12 words)
The solution to this conundrum, carried out first in Bishop's 1967 book, is to consider only functions which are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence. (40 words)
This is a consequence of the fact that, given two continuous functions : defined on the same domain I, then the sum f + g and the product fg of the two functions are continuous (on the same domain I). (38 words)
Distributions as derivatives of continuous functions The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of D(U) (or S(R d ) for tempered distributions). (35 words)
Example sentences (20)
Continuous functions are Borel functions but not all Borel functions are continuous.
Antiderivatives of non-continuous functions Non-continuous functions can have antiderivatives.
Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.
The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions.
This is a consequence of the fact that, given two continuous functions : defined on the same domain I, then the sum f + g and the product fg of the two functions are continuous (on the same domain I).
All uniformly continuous functions are continuous with respect to the induced topologies.
As an illustration, he examines Cauchy's proof that the sum of a series of continuous functions is itself continuous.
Continuous functions and homeomorphisms main A function or map from one topological space to another is called continuous if the inverse image of any open set is open.
For example, if a series of continuous functions converges uniformly, then the limit function is also continuous.
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.
When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit.
For instance where there are integral transforms in harmonic analysis for studying continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals.
In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. citation.
The solution to this conundrum, carried out first in Bishop's 1967 book, is to consider only functions which are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate of convergence.
A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.
Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions.
But if the maximum norm is used, the closed linear span will be the space of continuous functions on the interval.
Cited by citation Informally, this means that hardly any continuous functions have a derivative at even one point.
Distributions as derivatives of continuous functions The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of D(U) (or S(R d ) for tempered distributions).
Example: Decomposition of continuous functions Motivation for decomposition As to why the decomposition is valuable, the reason is twofold.