An Atlas of Epsilon-Delta Continuity Proofs in Function Space <em>F</em>(R, R)
Bibliographic record
Abstract
The ϵ-δ definition of continuity, foundational to real analysis, continues to attract sustained interest across the mathematics community. This paper responds to this interest by providing a comprehensive, systematic survey of direct ϵ-δ proofs for a broad spectrum of functions. We meticulously analyze 54 prominent real-valued functions, categorizing them into eight distinct clusters to highlight recurring proof structures and methodologies. For each function, we present a step-by-step proof alongside an explicit formula for δ in terms of ϵ and the point of continuity. Beyond serving as a robust pedagogical resource for students, instructors, and independent learners, this collection demystifies the proof-writing process by showcasing the elegant, unifying logic underlying a seemingly diverse set of problems. The work’s organized structure and detailed examples offer clarity where confusion often resides, ultimately fostering a deeper intuition for the core principles of continuity. By transforming a collection of challenging proofs into an accessible and navigable reference, this atlas opens up new avenues for further research in the field for mathematicians.
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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 | 0.011 | 0.005 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
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; both teacher heads agree on what is shown here.
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".