An Atlas of Epsilon-Delta Continuity Proofs in Function Space <em>F</em>(R, R)

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.