Quasi-continuous function

Key takeaways

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

Reference excerpt

In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general.

Definition Let X {\displaystyle X} be a topological space. A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is quasi-continuous at a point x ∈ X {\displaystyle x\in X} if for any ϵ > 0 {\displaystyle \epsilon >0} and any open neighborhood U {\displaystyle U} of x {\displaystyle x} there is a non-empty open set G ⊂ U {\displaystyle G\subset U} such that

| f ( x ) − f ( y ) | < ϵ ∀ y ∈ G {\displaystyle |f(x)-f(y)|<\epsilon \;\;\;\;\forall y\in G}

Note that in the above definition, it is not necessary that x ∈ G {\displaystyle x\in G} .

Properties If f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is continuous then f {\displaystyle f} is quasi-continuous If f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is continuous and g : X → R {\displaystyle g:X\rightarrow \mathbb {R} } is quasi-continuous, then f + g {\displaystyle f+g} is quasi-continuous.

Example Consider the function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } defined by f ( x ) = 0 {\displaystyle f(x)=0} whenever x ≤ 0 {\displaystyle x\leq 0} and f ( x ) = 1 {\displaystyle f(x)=1} whenever x > 0 {\displaystyle x>0} . Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set G ⊂ U {\displaystyle G\subset U} such that y < 0 ∀ y ∈ G {\displaystyle y<0\;\forall y\in G} . Clearly this yields | f ( 0 ) − f ( y ) | = 0 ∀ y ∈ G {\displaystyle |f(0)-f(y)|=0\;\forall y\in G} thus f is quasi-continuous. In contrast, the function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } defined by g ( x ) = 0 {\displaystyle g(x)=0} whenever x {\displaystyle x} is a rational number and g ( x ) = 1 {\displaystyle g(x)=1} whenever x {\displaystyle x} is an irrational number is nowhere quasi-continuous, since every nonempty open set G {\displaystyle G} contains some y 1 , y 2 {\displaystyle y_{1},y_{2}} with | g ( y 1 ) − g ( y 2 ) | = 1 {\displaystyle |g(y_{1})-g(y_{2})|=1} .

Worked examples

Example 1 — a first encounter with Quasi-continuous function

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

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

How to study Quasi-continuous function in 20 minutes

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

Frequently asked questions

What is Quasi-continuous function in simple terms?

In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general.

Why does Quasi-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 Quasi-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 Quasi-continuous function.