Uniform Continuity
δ depends only on ε, not the point.
Key idea
δ depends only on ε, not the point.
In depth
f is uniformly continuous if δ depends only on ε, not on the point. Continuous functions on compact intervals are uniformly continuous. f(x) = 1/x on (0, 1) is continuous but not uniformly so.
Worked example
f(x) = x² is not uniformly continuous on R since it steepens, but is on [0, 5].
Real-world uses
- Guaranteeing that a numerical method is stable across a whole interval, not just at one point
Common mistake
Assuming continuity on an open interval implies uniform continuity.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Uniform Continuity?
δ depends only on ε, not the point. It is a topic at the expert level in Analysis & Topology.
Where is Uniform Continuity used in real life?
Real-world examples: Guaranteeing that a numerical method is stable across a whole interval, not just at one point.
What should I learn before Uniform Continuity?
Make sure you are comfortable with Continuity (ε–δ), the previous topic in Analysis & Topology.
What is a common mistake with Uniform Continuity?
Assuming continuity on an open interval implies uniform continuity.
How can I practise Uniform Continuity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Uniform Continuity?
Move on to Extreme Value Theorem next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.