A generalization of Darboux-Froda theorem and its applications
Bibliographic record
Abstract
In real analysis, the Darboux-Froda theorem states that all discontinuities of a real-valued monotone functions of a real variable are at most countable. In this paper, we extend this theorem to <italic>a family</italic> of monotone <italic>real vector-valued functions</italic> of a real variable arising from dynamical systems. To this end, we explore some essential characteristics of countable and uncountable sets by the notions of strong cluster points, upper and lower strong cluster points, and establish the existence of strong cluster point sets, upper and lower strong cluster point sets for an uncountable set. With the help of these strong cluster point sets, we establish a jump lemma that helps characterize the discontinuities of the family of monotone vector-functions. Then we introduce the notion of distinction set and prove the existence of a distinction set. Making use of the upper and lower strong cluster points of the distinction set and the jump lemma, we prove the Darboux-Froda extension theorem. Moreover, we also present two applications of the generalized Darboux-Froda theorem.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".