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 with graphs.

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.