mathematics

Nowhere continuous function

Nowhere continuous 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 Nowhere continuous function rather than just read about it. In short: In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} there is some ε > 0 {\displaystyle \varepsilon >0} such that for every δ > 0 , {\displaystyle \delta…

Key takeaways

Nowhere continuous 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 Nowhere continuous function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Nowhere continuous function from memory before moving on to harder problems.

Reference excerpt

In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} there is some ε > 0 {\displaystyle \varepsilon >0} such that for every δ > 0 , {\displaystyle \delta >0,} we can find a point y {\displaystyle y} such that | x − y | < δ {\displaystyle |x-y|<\delta } and | f ( x ) − f ( y ) | ≥ ε {\displaystyle |f(x)-f(y)|\geq \varepsilon } . Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space.

Examples

Dirichlet function

One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} and has domain and codomain both equal to the real numbers. By definition, 1 Q ( x ) {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)} is equal to 1 {\displaystyle 1} if x {\displaystyle x} is a rational number and it is 0 {\displaystyle 0} otherwise. More generally, if E {\displaystyle E} is any subset of a topological space X {\displaystyle X} such that both E {\displaystyle E} and the complement of E {\displaystyle E} are dense in X , {\displaystyle X,} then the real-valued function which takes the value 1 {\displaystyle 1} on E {\displaystyle E} and 0 {\displaystyle 0} on the complement of E {\displaystyle E} will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.

Non-trivial additive functions

A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is called an additive function if it satisfies Cauchy's functional equation:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Nowhere continuous function

Start with the simplest possible case. Write down what Nowhere continuous 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 Nowhere continuous 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 Nowhere continuous 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 Nowhere continuous function

In research: Nowhere continuous 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 Nowhere continuous 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: Nowhere continuous function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical analysis, Topology, Types of functions, so understanding it makes those chapters shorter.
In everyday life: Look for Nowhere continuous 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Nowhere continuous function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Nowhere continuous 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 Nowhere continuous function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Nowhere continuous function in simple terms?

Why does Nowhere continuous 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 Nowhere continuous 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 Nowhere continuous function.