Differences in definitions of limit and continuity of functions as treated in courses on calculus and in rigorous undergraduate analysis yield contradictory outcomes and unexpected language. There are results about limits in calculus that are false by the definitions of analysis, functions not continuous by one definition and continuous by another, and continuous functions with discontinuities. This paper uncovers these issues and resolves them in five guided classroom discoveries ideal for students in a first course in real analysis, an honors calculus course, or a course on transitions to higher-level mathematical thinking. The conclusion provides definitions consistent with analysis in a setting for the general student of calculus.