Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- The real line, bounded intervals and rays are intervals, an interval contains the closed interval between two of its points, and every point of an interval with two distinct points is approached by other points of it.
A Continuous Injective Function on a Closed Interval is Strictly Monotone
problemprob:continuous-injective-strictly-monotone-2026aAnalysisAnalysis level. Injectivity plus continuity on a closed real interval forces strict monotonicity; the whole argument is repeated use of the intermediate value theorem.A Nonnegative Continuous Function with Zero Integral
problemprob:nonnegative-zero-integral-2026aAnalysisHonours to analysis level. A continuous nonnegative function on a closed interval whose integral vanishes is identically zero.A Function with Small Derivative Has Exactly One Fixed Point
problemprob:contraction-unique-fixed-point-2026aAnalysisHonours level. A function differentiable on the whole real line whose derivative is bounded in absolute value by one half has exactly one fixed point.- Honours level. Show that the sequence given by and is increasing and bounded above by , hence convergent, and that its limit is .
- Introductory to honours level. A continuous function on the unit interval taking equal values at the endpoints takes equal values at some pair of points half a unit apart.
- Introductory calculus. Show that has exactly one real zero and locate it in the open interval from to , combining the intermediate value theorem with strict monotonicity from the sign of the derivative.
The Rectangle of Greatest Area with a Given Perimeter
problemprob:rectangle-maximal-area-2026aAnalysisIntroductory calculus. Show that the area function of a rectangle of fixed perimeter attains a greatest value on the relevant closed interval and does so only at the square, using the extreme value theorem and the vanishing of the derivative at an interior extremum.A Limit Computed from the Epsilon-Delta Definition
problemprob:limit-square-epsilon-delta-2026aAnalysisBeginner level. Verify the limit of at straight from the epsilon-delta definition, by exhibiting a delta for each epsilon.The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds
lemmalem:derivative-sign-monotone-2026bAnalysisFor a function continuous on an interval and differentiable at the points strictly inside it, a nonnegative, positive, nonpositive, negative or zero derivative there makes the function nondecreasing, strictly increasing, nonincreasing, strictly decreasing or constant, and a bound…A Bounded Monotone Sequence of Real Numbers Converges
theoremthm:monotone-bounded-sequence-converges-2026aAnalysisA nondecreasing sequence of real numbers whose terms are bounded above converges to the supremum of its set of terms, and the nonincreasing case converges to the infimum.Intermediate Value Theorem on a Closed Real Interval, and Continuous Images of Intervals
theoremthm:intermediate-value-closed-interval-2026bAnalysisA real function continuous on a closed interval takes every value between its values at the endpoints, and the image of an interval under a continuous real function is an interval.Continuity at a Point of an Interval in Terms of the Limit
lemmalem:limit-continuity-bridge-2026aAnalysisA real function on an interval is continuous at a point of that interval exactly when its limit at the point exists and equals its value there.- Defines nondecreasing, nonincreasing, strictly increasing and strictly decreasing real-valued functions on a subset of the real line, and the terms monotone and strictly monotone.
- The epsilon-delta definition of the limit of a real-valued function at a point of an interval, and the notation for it. Uniqueness of the limiting value is established separately.
- Extends the real-line setting by carrying, by reference, the vocabulary and standard results of single-variable calculus: derivatives, the mean value theorem, uniform convergence, the Riemann integral, and the exponential and logarithm.
The Real Line: Standing Notation and Background for Calculus on Intervals
settingset:real-line-2026bAnalysisExtends the routine-facts setting with the notation and background for calculus on intervals of the real line: suprema, the real line and its subsets as metric spaces, intervals, continuity written out, square roots, and the compactness, Heine-Cantor, extreme value and intermedia…- Defines, from eight parameters, the data of a two-state controlled population of Ising type: the compact box of admissible transition rates, the rates themselves, the entropic running cost with its splitting interaction, the two observation channels, and the smooth extensions of…
Regimes of the Ising Population Model Outside the Fluctuation Optimality Theorem
remarkrem:ising-excluded-regimes-2026bProbabilityRecords which regimes of the Ising population model the fluctuation optimality theorem excludes and why: the ferromagnetic sign of the interaction breaks joint coercivity, a vanishing background observation rate breaks uniform observation positivity, and the convexity weight is a…The Ising Population Model: A Worked Instance of the Fluctuation Theory
exampleex:ising-population-model-2026aProbabilityA self-contained account of a two-state controlled population whose controls are the transition rates and whose observations are noisy per-state counts: its mean-field equilibrium, the scalar fluctuation problem it reduces to, and the closed-form asymptotically optimal value of i…