An Almost Everywhere Extension Theorem for Continuous Definable Functions in an O-minimal Structure
Bibliographic record
Abstract
Let $\mathcal{R} = (R, <, \mathcal{S})$ be an o-minimal expansion of an ordered group. In this thesis, we define the class $\mathcal{C}$ of asymptotically monotone cells and we show they have the property that, for any cell $C \in \mathcal{C}$ and for any definable, continuous, bounded function $f : C \rightarrow R$, it is always possible to continuously extend $f$ "almost everywhere" to the frontier of $C$. We make this notion precise using a theory of dimension for sets definable in an o-minimal structure. This result is a generalization of a known fact about continuous extensions of definable, continuous, bounded functions on open cells; we show by way of counterexample that the original result does not generalize to the class of all cells and hence that the assumption that our cells are asymptotically monotone is required. Background on o-minimality and the theory of dimension for definable sets is provided.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.027 | 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".