pointwise

USAGE SUMMARY

The phrase "pointwise" is correct and usable in written English.
It is typically used in mathematical or technical contexts to describe operations or comparisons made at individual points or instances rather than in a continuous manner. Example: "The function is evaluated pointwise to determine its behavior at specific values."

✓ Grammatically correct

Science

News & Media

Human-verified examples from authoritative sources

Exact Expressions

24 human-written examples

The ultimate solution is then obtained by gluing together the pointwise superpositions.

The Guardian

(As we saw above, Viète had leveled this charge against the instrumental, pointwise constructions provided by Clavius).

SEP

He maintains that these more complicated curves could be constructed by pointwise methods that offered greater precision, but the curves thus generated were no longer presented as absolutely geometrical.

SEP

As such, what Descartes suggests is that it is not possible to divide the arc any way we please, and we cannot therefore locate any arbitrary point along the curve by use of pointwise construction.

SEP

Figure 9 While consideration of Clavius's construction of the quadratrix offers some reason to accept Descartes' distinction between the different sorts of pointwise constructions, there remains the controversial claim that curves described by "generic" pointwise constructions are curves that can be constructed by continuous motion.

SEP

Finally, a ranking function ρ on A is regular just in case ρ(A) < ∞ for every non-empty or consistent proposition A in A. For more see Huber (2006), which discusses under which conditions ranking functions on fields of propositions induce pointwise ranking functions on the underlying set of possibilities.

SEP

(Note that UCT[0,1] is equivalent, relative to BISH, to the general uniform continuity theorem for metric spaces: Every pointwise continuous mapping of a complete, totally bounded metric space into a metric space is uniformly continuous. See Loeb 2005, and Bridges and Diener 2007).

SEP

His construction is a pointwise one: We begin with a quadrant of a circle (as in Pappus's description) but rather than relying on the intersection of uniformly moving segments to describe the curve, Clavius proceeds by first identifying the points of intersection between segments that bisect the quadrant and segments that bisect the arc of the quadrant.

SEP

According to Clavius's commentary of 1589, this pointwise construction of the quadratrix was an improvement over that offered by Pappus, because it was more accurate: Since the pointwise construction allowed one to identify arbitrarily many points along the curve, one could trace the quadratrix with greater precision than if one had to consider the intersection of two moving lines.

SEP

Based on this result, Descartes suggests a way to generalize further and solve the n-line Pappus Problem, for no matter how many given lines and angles with which a Pappus Problem begins, it will be possible to reduce the problem to an equation and then pointwise construct the roots of the equation, i.e., the sought after points C of the problem (G, 37).

SEP

The result of Descartes' analysis, as indicated by the remarks above, is that the curve that includes the sought after points C can be pointwise constructed by using ruler and compass to solve for the roots of a second-degree equation in two unknowns.

SEP

Expert writing Tips

Best practice

When using "pointwise", ensure the context clearly indicates that an operation or property applies to individual elements or points, rather than to a set as a whole. This is especially important in mathematical or technical writing.

Common error

Avoid using "pointwise" when describing properties that apply to an entire set or function collectively. "Pointwise" specifically denotes operations or characteristics considered at individual points.

Linguistic Context

Adjective/Adverb: "Pointwise" primarily functions as an adjective or adverb, modifying nouns or verbs to indicate that an action, property, or relationship applies to individual points within a set or domain. Ludwig examples show its use in describing mathematical constructions and properties like "pointwise superpositions".

Expression frequency: Common

Frequent in

Science

65%

Encyclopedias

15%

News & Media

10%

Less common in

Formal & Business

5%

Reference

3%

Wiki

2%

Ludwig's WRAP-UP

The term "pointwise" is primarily employed as an adjective or adverb within mathematical and technical contexts to denote operations or properties that apply individually to each element or point in a set, rather than to the set as a whole. Ludwig confirms its grammatical correctness and common use in formal and scientific writing. When clarity is paramount, alternatives such as "at each point" or "for each element" can enhance understanding. It's crucial to distinguish "pointwise" operations from those applied collectively to a set, ensuring precise communication in technical discourse.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

at each point

Focuses on individual points rather than the entire set.

for each element

Specifies application to individual elements within a set or domain.

element-wise

Highlights the operation's application to corresponding elements.

component by component

Indicates a detailed, element-by-element processing.

locally

Stresses the operation's confinement to small neighborhoods around each point.

at the individual level

Emphasizes a focus on singular, distinct instances.

separately for each

Underlines the distinct treatment of each element.

point-by-point

Indicates sequential processing at each point.

individually

Highlights separate and distinct treatment.

for every instance

Focuses on universal application across each occurrence.

FAQs

How is "pointwise" used in mathematical contexts?

In mathematics, "pointwise" describes operations or properties that apply to individual elements of a set or function. For example, "pointwise convergence" refers to the convergence of a function at each point in its domain.

What does "pointwise" mean in the context of function operations?

When applied to functions, "pointwise" means that an operation (like addition or multiplication) is performed on the function values at each point in the domain. This contrasts with operations that might consider the function as a whole.

Are there situations where using "at each point" is better than "pointwise"?

While "pointwise" is concise, "at each point" may provide better clarity in contexts where the individual application is not immediately obvious. The choice depends on the audience and the desired level of explicitness. Other alternatives include "element-wise" or "for each element".

What's the difference between "pointwise" and uniform convergence?

"Pointwise" convergence only requires that a sequence of functions converges at each point in the domain. Uniform convergence, however, requires that the convergence happens at the same rate across the entire domain. Uniform convergence is a stronger condition than "pointwise convergence".