Real Analysis glossary

Clear, one-line definitions of the Real Analysis terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

cauchy-mvt

Cauchy's mean value theorem: for f, g continuous on [a,b] and differentiable on (a,b), some c ∈ (a,b) satisfies [f(b)−f(a)]g'(c) = [g(b)−g(a)]f'(c).

continuous

A function f is continuous at a point a when lim_{x→a} f(x) = f(a); informally, f has no jumps, gaps, or blow-ups there.

See: Continuous Functions

continuous-at-point

f is continuous at a when for every ε > 0 there exists δ > 0 such that |x − a| < δ implies |f(x) − f(a)| < ε.

continuous-on-set

f is continuous on a set S when it is continuous at every point of S; uniform continuity is a stronger global condition.

delta

In ε-δ definitions, the positive number δ chosen (depending on ε) so that |x − a| < δ forces the output to stay within ε of the target.

epsilon

In ε-δ definitions, the positive tolerance ε on the output; for every such ε one must find a matching δ on the input.

essential-discontinuity

A discontinuity at a where at least one one-sided limit fails to exist (e.g., it oscillates or diverges to ±∞).

extreme-value-theorem

Every continuous function on a closed bounded interval [a, b] attains its maximum and minimum values on that interval.

intermediate-value-theorem

If f is continuous on [a, b] and y lies between f(a) and f(b), then f(c) = y for some c ∈ [a, b].

jump-discontinuity

A discontinuity at a where both one-sided limits exist but are unequal: lim_{x→a⁻} f(x) ≠ lim_{x→a⁺} f(x).

limit

The value f(x) approaches as x approaches a, written lim_{x→a} f(x); if the value depends on the path of approach, no limit exists.

lower-sum

For a partition P of [a, b], the sum L(f, P) = Σ mᵢ Δxᵢ where mᵢ is the infimum of f on the i-th subinterval.

one-sided-limit

The limit of f(x) as x approaches a from only one side, written lim_{x→a⁻} f(x) (left) or lim_{x→a⁺} f(x) (right).

pointwise-convergence

A sequence of functions fₙ converges pointwise to f when fₙ(x) → f(x) for each fixed x in the domain.

radius of convergence

For a power series Σ aₙ(x − c)ⁿ, the number R such that the series converges for |x − c| < R and diverges for |x − c| > R.

removable-discontinuity

A discontinuity at a where lim_{x→a} f(x) exists but differs from f(a) (or f(a) is undefined); fixable by redefining f(a).

riemann-integrable

A bounded function f on [a, b] is Riemann integrable when its upper and lower sums share a common limit as the partition refines.

riemann-integral

The common value ∫ₐᵇ f(x) dx of the upper and lower sums of f on [a, b], when that common value exists.

sup-norm-criterion

fₙ → f uniformly on S iff sup_{x ∈ S} |fₙ(x) − f(x)| → 0 as n → ∞; the supremum norm collapses uniform convergence to a numerical limit.

taylor series

The power series Σ f^(n)(a)(x − a)ⁿ / n! representing a smooth function f near a; converges to f within its radius of convergence.

taylor-remainder

The error Rₙ(x) = f(x) − Tₙ(x) between f and its degree-n Taylor polynomial; controlled by Lagrange or integral remainder formulas.

uniform-convergence

fₙ → f uniformly on S when for every ε > 0 there is N such that |fₙ(x) − f(x)| < ε for all n ≥ N and all x ∈ S.

upper-sum

For a partition P of [a, b], the sum U(f, P) = Σ Mᵢ Δxᵢ where Mᵢ is the supremum of f on the i-th subinterval.

weierstrass-m-test

If |fₙ(x)| ≤ Mₙ on S and Σ Mₙ converges, then Σ fₙ(x) converges uniformly (and absolutely) on S.