a smooth function

USAGE SUMMARY

The phrase "a smooth function" is correct and usable in written English.
It is typically used in mathematics and analysis to describe a function that is continuously differentiable, meaning it has derivatives of all orders. Example: "In calculus, we often work with a smooth function to ensure that we can apply various theorems and techniques effectively."

✓ Grammatically correct

Science

News & Media

Academia

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

The switching function has been designed as a smooth function.

Chaos, Solitons & Fractals

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

The interface is identified as the zero level set of a smooth function.

Journal of Computational Physics

The first key observation is the following lemma which shows that a smooth function (not necessarily convex) is 'sandwiched' between two quadratic functions.

Princeton University

In the control design, a smooth function is introduced with backstepping technique to compensate for the effects of interactions.

IFAC-PapersOnLine

We only assume that the global contribution received by each scene point is a smooth function with respect to the frequency of the lighting.

Columbia University

Let u ( x ) be a smooth function.

Boundary Value Problems

Suppose that ϕ is a smooth function.

Journal of Inequalities and Applications

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

Expert writing Tips

Best practice

When using "a smooth function" in mathematical writing, define the level of smoothness required for your specific application (e.g., C^∞ for infinitely differentiable).

Common error

Avoid assuming that a continuous function is automatically "a smooth function". Continuity only means there are no breaks, while smoothness requires continuous differentiability.

Linguistic Context

The phrase "a smooth function" functions primarily as a descriptor in mathematical contexts, identifying a function that is continuously differentiable to all orders. As Ludwig AI confirms, this phrase is grammatically correct and common in academic and scientific texts.

Expression frequency: Very common

Frequent in

Science

89%

Academia

7%

News & Media

1%

Less common in

Formal & Business

1%

Encyclopedias

0%

Wiki

0%

Ludwig's WRAP-UP

In summary, "a smooth function" is a mathematical term referring to a function with continuous derivatives of all orders. Ludwig AI confirms its grammatical correctness, and it's commonly used in scientific and academic fields to describe functions with specific analytical properties. When writing, ensure you understand the level of smoothness needed and avoid confusing continuity with differentiability. Alternatives like "continuously differentiable function" can clarify your meaning. By understanding these nuances, you can use "a smooth function" effectively in your writing.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

a continuously differentiable function

Specifies a key characteristic of smoothness in mathematical terms.

a function with continuous derivatives

Emphasizes the derivative property related to smoothness.

a well-behaved function

Describes a function that is predictable and easy to work with, similar to being smooth.

a regular function

Indicates a function that meets certain regularity conditions, relating to smoothness.

a gentle function

Uses an analogy to describe the gradual and predictable change in the function's values.

a polished function

Suggests that the function has been refined to eliminate irregularities, similar to smoothness.

a predictable function

Highlights the consistent and non-erratic behavior of the function, akin to smoothness.

a flowing function

Uses a metaphor to depict the seamless and uninterrupted nature of the function.

a seamless function

Focuses on the continuous and unbroken nature of the function's curve, representing smoothness.

a graceful function

Implies elegance and lack of abrupt changes in the function's behavior, similar to smoothness.

FAQs

How is "a smooth function" typically used in mathematics?

In mathematics, "a smooth function" is used to describe a function that possesses derivatives of all orders. This property is essential for many theorems and techniques in calculus and analysis.

What are some properties of "a smooth function"?

Smooth functions are characterized by their infinite differentiability. This means they have derivatives of all orders, and these derivatives are continuous. Examples of functions sharing some properties include "differentiable functions" or "continuous functions".

In what fields besides mathematics is the concept of "a smooth function" relevant?

The concept extends to physics, engineering, and computer graphics, where smooth functions are used to model physical phenomena, design curves and surfaces, and ensure stable numerical computations.

What's the difference between "a smooth function" and a continuous function?

A continuous function has no breaks or jumps in its graph, but it may have sharp corners or kinks. A "a smooth function" not only has no breaks but also has continuous derivatives of all orders, meaning its graph is infinitely differentiable.