The definitions of a limit and continuity at a point for a function usually appear in a calculus
text after numerical motivation using tables and some graphs.
A related collection of definitions for one sided limits and continuity
usually appear directly after the traditional definitions to allow
further consideration of functions defined by cases and functions with
domains other than open intervals.
Here are traditional versions of these definitions.
Definition of Limit: CCD.DL.
Definition of One-Sided Limit: CCD.DOSL
Definition of Continuity (at a Point): CCD.DCP
Definition of Continuity (on an Interval): CCD.DCI
Dynamic visualization of the definitions of limit and continuity:
Examples of Limits and Continuity with Mapping Diagrams and Graphs.
The connection between limits and continuity are reinforced with several
key illuminating examples, improved by using mapping diagrams along
CCD.DLC.Example.1. A Removeable Discontinuity.
CCD.DLC.Example.2. A Step Discontinuity.
CCD.DLC.Example.3. A Vertical Asymptote Discontinuity ("Blowing Up").
CCD.DLC.Example.4. Mathematician's Sine Wave's.
CCD.DLC.Example.5. Discontinuities Almost Everywhere.