mathematics

Integer-valued function

Integer-valued function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Integer-valued function rather than just read about it. In short: In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain.

Key takeaways

Integer-valued function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Integer-valued function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Integer-valued function from memory before moving on to harder problems.

Reference excerpt

In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain. The floor and ceiling functions are examples of integer-valued functions of a real variable, but on real numbers and, generally, on (non-disconnected) topological spaces integer-valued functions are not especially useful. Any such function on a connected space either has discontinuities or is constant. On the other hand, on discrete and other totally disconnected spaces integer-valued functions have roughly the same importance as real-valued functions have on non-discrete spaces. Any function with natural, or non-negative integer values is a partial case of an integer-valued function.

Examples Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, the Dirichlet function, the sign function and the Heaviside step function (except possibly at 0). Integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting function.

Algebraic properties On an arbitrary set X, integer-valued functions form a ring with pointwise operations of addition and multiplication, and also an algebra over the ring Z of integers. Since the latter is an ordered ring, the functions form a partially ordered ring:

f ≤ g ⟺ ∀ x : f ( x ) ≤ g ( x ) . {\displaystyle f\leq g\quad \iff \quad \forall x:f(x)\leq g(x).}

Uses

Graph theory and algebra Integer-valued functions are ubiquitous in graph theory. They also have similar uses in geometric group theory, where length function represents the concept of norm, and word metric represents the concept of metric. Integer-valued polynomials are important in ring theory.

Mathematical logic and computability theory In mathematical logic, such concepts as primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions on Nn. Gödel numbering, defined on well-formed formulae of some formal language, is a natural-valued function. Computability theory is essentially based on natural numbers and natural (or integer) functions on them.

Number theory In number theory, many arithmetic functions are integer-valued.

Computer science In computer programming, many functions return values of integer type due to simplicity of implementation.

See also Integer-valued polynomial Semi-continuity Rank (disambiguation)#Mathematics Grade (disambiguation)#In mathematics

References

Further reading https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf

Illustrations

Worked examples

Example 1 — a first encounter with Integer-valued function

Start with the simplest possible case. Write down what Integer-valued function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Integer-valued function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Integer-valued function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Integer-valued function

In research: Integer-valued function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Integer-valued function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Integer-valued function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Types of functions, so understanding it makes those chapters shorter.
In everyday life: Look for Integer-valued function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Integer-valued function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Integer-valued function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Integer-valued function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Integer-valued function in simple terms?

In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain.

Why does Integer-valued function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Integer-valued function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Integer-valued function.