Continuity (ε–δ)
Formal continuity at a point.
Key idea
Formal continuity at a point.
In depth
f is continuous at a if for each ε there is δ with |x − a| < δ implying |f(x) − f(a)| < ε. Unlike a limit, the point x = a is included and f(a) must be defined.
Worked example
f(x) = x² is continuous at 3: take δ = min(1, ε/7).
Real-world uses
- Formalising why small input changes give small output changes in control systems
Common mistake
Using a limit definition that excludes x = a.
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 Continuity (ε–δ)?
Formal continuity at a point. It is a topic at the expert level in Analysis & Topology.
Where is Continuity (ε–δ) used in real life?
Real-world examples: Formalising why small input changes give small output changes in control systems.
What should I learn before Continuity (ε–δ)?
Make sure you are comfortable with Limits of Functions (ε–δ), the previous topic in Analysis & Topology.
What is a common mistake with Continuity (ε–δ)?
Using a limit definition that excludes x = a.
How can I practise Continuity (ε–δ)?
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 Continuity (ε–δ)?
Move on to Uniform Continuity 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.