Skip to content

Real Analysis

← Back to Domain-Specific Abstractions by Domain

11 domain-specific abstractions whose origin domain is Real Analysis.

  • Absolute continuity — A strengthened continuity property that makes total function variation over sufficiently short disjoint intervals arbitrarily small and restores integration of the derivative.
  • Baire one star function — A real function whose restriction is continuous on some nonempty relatively open piece of every perfect set.
  • Helly's selection theorem — Every uniformly bounded sequence of monotone real functions on a compact interval has a subsequence converging pointwise, with bounded-variation generalizations providing compactness for measure and weak-convergence arguments.
  • Locally integrable function — A measurable function whose absolute value has finite integral on every compact subset of its open domain, without requiring finite integral over the whole domain.
  • McShane integral — A gauge integral using free tagged partitions whose tags may lie outside their associated subintervals, yielding for real-valued functions exactly the Lebesgue-integrable class.
  • Nowhere continuous function — A function that fails the continuity condition at every point of its domain.
  • Riemann sum — Approximate a definite integral by partitioning an interval, sampling the function once in each subinterval, multiplying each sampled value by subinterval width, and summing, with mesh refinement controlling convergence.
  • Rising sun lemma — A one-dimensional decomposition lemma partitioning the points below a later higher value of a continuous function into disjoint intervals with controlled endpoints.
  • Singular function — A nonconstant continuous real function whose derivative exists and equals zero almost everywhere.
  • Symmetrically continuous function — A real function whose values at equally spaced points on opposite sides of each point approach one another.
  • Weierstrass function — A classical infinite trigonometric series that is continuous everywhere and differentiable nowhere under suitable parameters.