Integrability Criteria
Continuous or monotone functions are integrable.
Key idea
Continuous or monotone functions are integrable.
In depth
f is integrable iff for every ε there is a partition with U(f, P) − L(f, P) < ε. Continuous functions and monotone functions are integrable. By Lebesgue’s criterion, a bounded f is integrable iff its discontinuities have measure zero.
Worked example
The Dirichlet function (1 on rationals, 0 on irrationals) is not Riemann integrable.
Real-world uses
- Knowing which measured functions can safely be integrated in engineering software
Common mistake
Assuming a function with infinitely many discontinuities is never integrable.
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 Integrability Criteria?
Continuous or monotone functions are integrable. It is a topic at the expert level in Analysis & Topology.
Where is Integrability Criteria used in real life?
Real-world examples: Knowing which measured functions can safely be integrated in engineering software.
What should I learn before Integrability Criteria?
Make sure you are comfortable with Riemann Integral, the previous topic in Analysis & Topology.
What is a common mistake with Integrability Criteria?
Assuming a function with infinitely many discontinuities is never integrable.
How can I practise Integrability Criteria?
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 Integrability Criteria?
Move on to Pointwise vs Uniform Convergence 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.