continuity of g

USAGE SUMMARY

The phrase "continuity of g" is correct and usable in written English.
It can be used in mathematical or technical contexts when discussing the continuous nature of a function or variable denoted by "g." Example: "In our analysis, we need to ensure the continuity of g to maintain the integrity of our results."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

53 human-written examples

The same conclusion is found when g is assumed to be continuous since continuity of g implies continuity of f.

Fixed Point Theory and Applications

If the conditions of Lemma 3.2 are strengthened, then we can get the continuity of g.

Journal of Inequalities and Applications

By the continuity of g at x and y, we get.

Fixed Point Theory and Applications

Continuity of g implies p x,Sx leq gbigl 0,p x,Sx),0,0+p x,Sx -0,0bigr).

Fixed Point Theory and Applications

By the continuity of g at x and y, we get x = g x ∈ ( F x ) α.

Fixed Point Theory and Applications

If (c) holds, then, by the continuity of g at x, we obtain that (x=gxin T x)).

Fixed Point Theory and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

7 human-written examples

Theorem 3.3 can be proved without assuming the b-continuity of f or the b-continuity of g.

Fixed Point Theory and Applications

By the lower semi-continuity of G and Lemma 3.1 [9], the mapping y ↦ ⋃ x ∈ X G ( x, y ) is lower semi-continuous with nonempty compact values.

Journal of Inequalities and Applications

The proof method of the lower semi-continuity of -g is similar to that of the upper semi-continuity of g and so the proof is omitted.

Journal of Inequalities and Applications

By the lower semi-continuity of g, we have gbigl(x_{0}, y'bigr)leqvarepsilon.

Journal of Inequalities and Applications

Using the G-continuity of g, Definition 1.8 and Lemma 1.4, we have lim n → ∞ G ( g g x n, g g x n, g x ) = lim n → ∞ G ( g g x n, g x, g x ) = 0, (2.24).

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When discussing mathematical proofs or conditions, ensure that the context clearly defines what 'g' represents to avoid ambiguity. For example, "Given a function g, the continuity of g is essential for..."

Common error

Avoid assuming that "continuity of g" automatically implies differentiability. A function can be continuous without being differentiable at every point. Make sure to differentiate the two concepts clearly in your writing.

Linguistic Context

The phrase "continuity of g" functions as a noun phrase, typically serving as the subject or object of a sentence. It refers to the property of a mathematical function 'g' being continuous. Ludwig AI indicates the phrase is correct and usable in written English, mainly within technical contexts.

Expression frequency: Common

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Academia

0%

Ludwig's WRAP-UP

The phrase "continuity of g" is a grammatically sound and commonly employed term, particularly within mathematical and scientific domains. As supported by Ludwig, its purpose is to denote the continuous nature of a function 'g', a critical property in various analytical contexts. While straightforward in its application, it's essential to distinguish continuity from related concepts like differentiability. Variations such as "g's continuity" or "the continuous nature of g" offer alternative phrasing. Remember, while useful to describe that property of "g", its usage is tied to formal and scientific contexts, given the preponderance of scientific sources listed by Ludwig.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

g's continuity

This alternative uses a possessive form to indicate the continuity of the function g.

the continuous nature of g

This phrase emphasizes the nature of g being continuous.

g being continuous

This option uses a gerund to express the continuity of g.

g is continuous

A simple statement declaring that g is continuous.

the property of g being continuous

This version highlights the property or attribute of g's continuity.

the continuous behavior of g

This alternative focuses on the behavioral aspect of continuity.

g remains continuous

This implies that g maintains its continuous state over a period or range.

the unbrokenness of g

This uses "unbrokenness" as a metaphor for continuity.

maintenance of g's continuity

This emphasizes the preservation or upkeep of g's continuity.

the sustained continuity of g

Adding 'sustained' highlights that the continuity is maintained over time.

FAQs

How does "continuity of g" affect mathematical models?

The "continuity of g" is often a necessary condition for many theorems and proofs in calculus and analysis. Without "continuity of g", certain mathematical operations and deductions may not be valid, affecting the accuracy and reliability of the model.

What is the difference between "continuity of g" and differentiability of g?

"Continuity of g" means that there are no breaks or jumps in the graph of the function, while differentiability requires that the function has a derivative at every point. A function can be continuous but not differentiable, but differentiability implies continuity.

In what contexts is "continuity of g" most relevant?

The "continuity of g" is especially important in fields such as real analysis, topology, and numerical analysis, where the behavior of functions under limits and transformations is critical. It is also crucial in physics and engineering, where continuous models are often used to describe real-world phenomena.

What are some synonyms for the term "continuity of g"?

While the phrase "continuity of g" is quite specific, you could rephrase it as "g's continuous nature" or "g being continuous" depending on the context. These alternatives emphasize the property of g being continuous.