be a smooth function

USAGE SUMMARY

The phrase "be a smooth function" is correct and usable in written English.
It is typically used in mathematical contexts to describe a function that is continuously differentiable, meaning it has derivatives of all orders. Example: "For the analysis to hold, we need to ensure that the function we are working with will be a smooth function across the entire interval."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

20 human-written examples

Let u ( x ) be a smooth function.

Boundary Value Problems

Let (z(t)inmathbb {R}) be a smooth function.

Advances in Difference Equations

Let f be a smooth function with compact support.

Boundary Value Problems

Corollary 1 Let u ( x ) be a smooth function.

Boundary Value Problems

Lemma 1 Let u ( x ) be a smooth function.

Boundary Value Problems

Let,, and let be a smooth function such that for each the derivative is surjective.

Advances in Difference Equations

Human-verified similar examples from authoritative sources

Similar Expressions

40 human-written examples

If the solution were a smooth function, one could carry out a Taylor expansion, which is a way of approximating the function by polynomials of increasingly higher degree.

The Guardian

where is a smooth function.

Boundary Value Problems

Suppose that ϕ is a smooth function.

Journal of Inequalities and Applications

The Green's function constructed below is a smooth function.

Boundary Value Problems

It is obvious that is a smooth function.

Advances in Difference Equations

Expert writing Tips

Best practice

When describing mathematical models, use "be a smooth function" to indicate that the function has continuous derivatives of all orders, ensuring predictable behavior in calculations and analyses.

Common error

Do not assume that a continuous function is necessarily a smooth function. Continuity only requires that there are no breaks in the function, while smoothness requires continuous differentiability. Ensure the function meets the differentiability criteria before stating it "be a smooth function".

Linguistic Context

The phrase "be a smooth function" serves as a predicate in mathematical statements, attributing the property of smoothness (infinite differentiability) to a function. As Ludwig AI points out, this implies that the function possesses continuous derivatives of all orders.

Expression frequency: Common

Frequent in

Science

95%

Formal & Business

2%

News & Media

1%

Less common in

Academia

1%

Encyclopedias

0%

Wiki

0%

Ludwig's WRAP-UP

In summary, "be a smooth function" is a phrase used primarily in mathematics and related scientific fields to denote a function with continuous derivatives of all orders. According to Ludwig AI, the phrase is grammatically correct and appears "Common"ly in academic and scientific publications. When using this phrase, ensure that the function indeed satisfies the condition of infinite differentiability. While continuity is a necessary condition, it is not sufficient. Consider alternatives like "infinitely differentiable" if greater precision is required.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

be a continuously differentiable function

Adds emphasis to the continuous differentiability of the function.

be a function of class C∞

Uses mathematical notation to denote infinite differentiability.

be infinitely differentiable

More technically specific, explicitly stating infinite differentiability.

have continuous derivatives

Emphasizes the continuity of the derivatives.

exhibit smoothness

Focuses on the property of smoothness itself, rather than defining the function.

be a regular function

Describes a function that is smooth and without singularities.

possess differentiability

Highlights the differentiability aspect, implying smoothness.

display smooth behavior

Shifts the focus to the function's behavior rather than its formal definition.

have a smooth profile

Focuses on the graphical representation of the function.

be well-behaved

Informal term indicating desirable mathematical properties, including smoothness.

FAQs

What does it mean for a function to "be a smooth function"?

For a function to "be a smooth function", it means that it has derivatives of all orders and these derivatives are continuous. In simpler terms, the function is infinitely differentiable.

How can I determine if a function /s/is+differentiable is also a smooth function?

To determine if a differentiable function is also a smooth function, you need to check if all its derivatives exist and are continuous. If you can differentiate the function infinitely many times without encountering any discontinuities, then it is a smooth function.

What are some examples of functions that "be a smooth function"?

Examples of functions that "be a smooth function" include polynomials, exponential functions, sine and cosine functions, and Gaussian functions. These functions have continuous derivatives of all orders.

In what contexts is the term "be a smooth function" typically used?

The term "be a smooth function" is commonly used in mathematics, physics, engineering, and computer science, particularly in areas such as calculus, differential equations, signal processing, and machine learning where the differentiability properties of functions are important.