找到全部一致连续的整函数并证明

Ijikelezisa 1 ye 40 · Paused

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Injongo

找到全部一致连续的整函数并证明

Ibar

每一个步骤都合理,每个假设都说明,没有关于可信或直觉的申诉。

Indawo Yolawulo

A Thinking Machines Inkling (free)

B NVIDIA Nemotron 3 Ultra (free)

Ref NVIDIA Nemotron 3 Ultra (free)

Iinjongo

open

A precise statement of the theorem is given: the set of all entire functions f: ℂ → ℂ that are uniformly continuous on ℂ is exactly { f(z) = az + b | a, b ∈ ℂ }.

open

Definitions of 'entire function' and 'uniformly continuous on ℂ' are explicitly stated and used consistently throughout the proof.

open

Proof that every function of the form f(z) = az + b (a, b ∈ ℂ) is entire and uniformly continuous on ℂ, with explicit ε-δ argument for uniform continuity.

open

Proof that if f is entire and uniformly continuous on ℂ, then f must be a polynomial of degree at most 1, using Cauchy's estimates or Liouville's theorem on the derivative, with all estimates fully justified and no appeal to intuition.

open

Proof that no entire function of degree ≥ 2 (e.g., z², e^z, sin z) is uniformly continuous on ℂ, by exhibiting explicit sequences violating the uniform continuity condition or by growth-rate arguments with explicit constants.

open

The complete argument is self-contained: every lemma invoked (Cauchy integral formula, Liouville's theorem, etc.) is either proved in-line or cited with a precise reference, all quantifiers are explicit, and no step relies on 'it is clear that', 'obviously', or similar gaps.

Ushicilelo oluneenkcukacha

Ref ijikelezayo 0 · System

Acceptance criteria set:
1. A precise statement of the theorem is given: the set of all entire functions f: ℂ → ℂ that are uniformly continuous on ℂ is exactly { f(z) = az + b | a, b ∈ ℂ }.
2. Definitions of 'entire function' and 'uniformly continuous on ℂ' are explicitly stated and used consistently throughout the proof.
3. Proof that every function of the form f(z) = az + b (a, b ∈ ℂ) is entire and uniformly continuous on ℂ, with explicit ε-δ argument for uniform continuity.
4. Proof that if f is entire and uniformly continuous on ℂ, then f must be a polynomial of degree at most 1, using Cauchy's estimates or Liouville's theorem on the derivative, with all estimates fully justified and no appeal to intuition.
5. Proof that no entire function of degree ≥ 2 (e.g., z², e^z, sin z) is uniformly continuous on ℂ, by exhibiting explicit sequences violating the uniform continuity condition or by growth-rate arguments with explicit constants.
6. The complete argument is self-contained: every lemma invoked (Cauchy integral formula, Liouville's theorem, etc.) is either proved in-line or cited with a precise reference, all quantifiers are explicit, and no step relies on 'it is clear that', 'obviously', or similar gaps.