Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The exponential is a continuous bijection from onto
Statement
is a bijection.
Facts & Assumptions
Given: The real exponential function.
It is strictly increasing (The exponential function is strictly increasing) and continuous (The sum of a real power series is continuous at every point strictly inside its interval of convergence).
Its values are positive, and its limits at the two ends are and (The exponential tends to at and to at ).
A continuous function on an interval takes every intermediate value (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on takes every value between and ).
Proof
Strict increase gives injectivity.
Given , [L2] provides with . Applying [L3] on gives with .
Positivity gives the stated codomain, and steps 1.1 and 1.2 give bijectivity.
Depends on
- The exponential function is strictly increasing
- The exponential tends to $+\infty$ at $+\infty$ and to $0$ at $-\infty$
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$
- The sum of a real power series is continuous at every point strictly inside its interval of convergence
Used by
- Assuming choice, a Hamel-basis additive map transported through exp gives a discontinuous logarithmic function that is not c log Counterexample
- The exponential map is a holomorphic surjection C to C^× that is not an automorphism Counterexample
- The natural logarithm as the inverse of the exponential function Definition
- A value outside the image can still be regular Example
- A regular value need not belong to the image False statement
- Zero free entire function of exponential type is an exponential Lemma
- The exponential roadmap and its circularity hazards Remark
- Standard Maclaurin expansions Theorem
Dependency tree · two levels
29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)