Uniform continuity

Key takeaways

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

Reference excerpt

In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number ε {\displaystyle \varepsilon } , then there is a positive real number δ {\displaystyle \delta } such that | f ( x ) − f ( y ) | < ε {\displaystyle |f(x)-f(y)|<\varepsilon } for any x {\displaystyle x} and y {\displaystyle y} in any interval of length δ {\displaystyle \delta } within the domain of f {\displaystyle f} . The difference between uniform continuity and (ordinary) continuity is that in uniform continuity there is a globally applicable δ {\displaystyle \delta } (the size of a function domain interval over which function value differences are less than ε {\displaystyle \varepsilon } ) that depends on only ε {\displaystyle \varepsilon } , while in (ordinary) continuity there is a locally applicable δ {\displaystyle \delta } that depends on both ε {\displaystyle \varepsilon } and x {\displaystyle x} . So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces. Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as f ( x ) = 1 x {\displaystyle f(x)={\tfrac {1}{x}}} on ( 0 , 1 ) {\displaystyle (0,1)} , or if their slopes become unbounded on an infinite domain, such as f ( x ) = x 2 {\displaystyle f(x)=x^{2}} on the real (number) line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map). Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.

Definition for functions on metric spaces For a function f : X → Y {\displaystyle f:X\to Y} with metric spaces ( X , d 1 ) {\displaystyle (X,d_{1})} and ( Y , d 2 ) {\displaystyle (Y,d_{2})} , the following definitions of uniform continuity and (ordinary) continuity hold.

Definition of uniform continuity

… excerpt ends here. Continue reading the full article.

Illustrations

Uniform continuity: As the center of the blue window, with real height

2
ε
∈

R

>
0

{\displaystyle 2\varepsilon \in \mathbb {R} _{>0}}

and real width

2
δ
∈

R

>
0

{\displaystyle 2\delta \in \mathbb {R} _{>0}}

, moves over the graph of

f
(
x
)
=

1
x

{\displaystyle f(x)={\tfrac {1}{x}}}

in the direction of

x
=
0

{\displaystyle x=0}

, there comes a point at which the graph of

f

{\displaystyle f}

penetrates the (interior of the) top and/or bottom of that window. This means that

f

{\displaystyle f}

ranges over an interval larger than or equal to

ε

{\displaystyle \varepsilon }

over an

x

{\displaystyle x}

-interval smaller than

δ

{\displaystyle \delta }

. If there existed a window whereof top and/or bottom is never penetrated by the graph of

f

{\displaystyle f}

as the window moves along it over its domain, then that window's width would need to be infinitesimally small (nonreal), meaning that

f
(
x
)

{\displaystyle f(x)}

is not uniformly continuous. The function

g
(
x
)
=

x

{\displaystyle g(x)={\sqrt {x}}}

, on the other hand, is uniformly continuous. — As the center of the blue window, with real height 2 ε ∈ R > 0 {\displaystyle 2\varepsilon \in \mathbb {R} _{>0}} and real width 2 δ ∈ R > 0 {\displaystyle 2\delta \in \mathbb {R} _{>0}} , moves over the graph of f ( x ) = 1 x {\displaystyle f(x)={\tfrac {1}{x}}} in the direction of x = 0 {\displaystyle x=0} , there comes a point at which the graph of f {\displaystyle f} penetrates the (interior of the) top and/or bottom of that window. This means that f {\displaystyle f} ranges over an interval larger than or equal to ε {\displaystyle \varepsilon } over an x {\displaystyle x} -interval smaller than δ {\displaystyle \delta } . If there existed a window whereof top and/or bottom is never penetrated by the graph of f {\displaystyle f} as the window moves along it over its domain, then that window's width would need to be infinitesimally small (nonreal), meaning that f ( x ) {\displaystyle f(x)} is not uniformly continuous. The function g ( x ) = x {\displaystyle g(x)={\sqrt {x}}} , on the other hand, is uniformly continuous.

Worked examples

Example 1 — a first encounter with Uniform continuity

Start with the simplest possible case. Write down what Uniform continuity 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 Uniform continuity 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 Uniform continuity 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 Uniform continuity

In research: Uniform continuity 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 Uniform continuity 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: Uniform continuity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Calculus, General topology, Mathematical analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Uniform continuity 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 Uniform continuity in 20 minutes

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

Frequently asked questions

What is Uniform continuity in simple terms?

In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are as close to each other as we want…

Why does Uniform continuity 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 Uniform continuity?

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 Uniform continuity.