Principles of Mathematical Analysis¶
Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
Cited by¶
11 citations across 11 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Boundedness
- Boundedness is the structural property that the values, magnitudes, or resource-uses of a set, sequence, function, or process do not exceed some fixed finite threshold — formally, a subset \(S\) of a metric space \((X, d)\) is bounded if there exist a centre \(x_0 \in X\) and a finite radius \(M < \infty\) such that \(d(x, x_0) \leq M\) for every \(x \in S\), with the equivalent formulation that \(S\) has finite diameter \(\operatorname{diam}(S) := \sup\{d(x, y) : x, y \in S\} < \infty\), as Rudin (1976) develops in his standard treatment of metric spaces.
This sourcestandard treatment of metric spaces in which a subset is bounded iff it is contained in a ball of finite radius / has finite diameter.
- Boundedness is the structural property that the values, magnitudes, or resource-uses of a set, sequence, function, or process do not exceed some fixed finite threshold — formally, a subset \(S\) of a metric space \((X, d)\) is bounded if there exist a centre \(x_0 \in X\) and a finite radius \(M < \infty\) such that \(d(x, x_0) \leq M\) for every \(x \in S\), with the equivalent formulation that \(S\) has finite diameter \(\operatorname{diam}(S) := \sup\{d(x, y) : x, y \in S\} < \infty\), as Rudin (1976) develops in his standard treatment of metric spaces.
- Continuity
- 4. Continuity property and its scope: the variant of continuity is named — pointwise continuity at
x₀(the epsilon-delta condition holds at one point), continuity onX(it holds at every point), uniform continuity onX— Rudin (1976)This sourceCanonical undergraduate real-analysis textbook; develops uniform continuity, the classification of discontinuities, the Weierstrass extreme-value theorem, and the Bolzano-Weierstrass theorem within the standard epsilon-delta framework.
- 4. Continuity property and its scope: the variant of continuity is named — pointwise continuity at
- Dense Set
- A is dense in B (with respect to a notion of closeness) exactly when the closure of A equals B
This sourceStandard reference defining density (closure equals the host), the density and countability of the rationals in the uncountable reals, and the ε-neighbourhood characterization.
- A is dense in B (with respect to a notion of closeness) exactly when the closure of A equals B
- Enthymeme
- Consider a proof that writes "since \(f\) is continuous on the compact set \(K\), it attains its maximum" and stops.
This sourceStandard analysis text establishing the extreme value theorem (a continuous function on a compact set attains its maximum), the suppressed premise in the worked proof example.
- Consider a proof that writes "since \(f\) is continuous on the compact set \(K\), it attains its maximum" and stops.
- Higher Order Function
- The composition algebra over rules is fully present and load-bearing: \(D\) composes with itself (\(D^2\) is the second derivative), has an approximate inverse (integration, via the fundamental theorem of calculus), and obeys distributive-style laws (linearity: \(D(af + bg) = aD(f) + bD(g)\); the product and chain rules govern composition).
This sourceEstablishes linearity of the derivative, the product and chain rules, and the fundamental theorem of calculus relating differentiation and integration — the composition algebra over the derivative operator.
- The composition algebra over rules is fully present and load-bearing: \(D\) composes with itself (\(D^2\) is the second derivative), has an approximate inverse (integration, via the fundamental theorem of calculus), and obeys distributive-style laws (linearity: \(D(af + bg) = aD(f) + bD(g)\); the product and chain rules govern composition).
- Quantifier
- Lift \(\delta\) outside the \(\forall x\) but let it still depend on \(\varepsilon\) alone across the whole domain, and you get uniform continuity — a strictly stronger claim, satisfied by fewer functions.
This sourceStandard source for the ε–δ definition of a limit and for the pointwise-versus-uniform continuity distinction, which turns entirely on quantifier order.
- Lift \(\delta\) outside the \(\forall x\) but let it still depend on \(\varepsilon\) alone across the whole domain, and you get uniform continuity — a strictly stronger claim, satisfied by fewer functions.
Domain-specific¶
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