Continuous Function: A function is continuous if its graph can be drawn without lifting the pencil off the paper; it is a smooth and connected curve.
Mnemonic: Trace a continuous curve without lifting the pencil.
Discontinuous Function: A function with breaks or gaps in its graph.
Points of Discontinuity: Locations on the x-axis where the function is not continuous.
Types of Discontinuity:
Jump Discontinuity: Function graph jumps from one point to another.
Removable Discontinuity: (To be covered in another video)
Infinite Discontinuity: (To be covered in another video)
Visualizing Discontinuity
Gaps/Breaks: Graphs of discontinuous functions have gaps or breaks which are called points of discontinuity.
Filled and Unfilled Circles:
Filled Circles: Indicate that the function is defined at that point.
Unfilled Circles (Holes): Indicate breaks; the function is not defined at that point.
Important Observations
Function Domain: Continuity and discontinuity are defined only within the domain of the function.
Example: A function undefined between x = 2 and x = 4 is not considered discontinuous in that interval since those points are outside the function's domain.
Example Graph Analysis
Shows a function with jump discontinuities at certain x-values (e.g., x3).
Graphs not standard between points where the function is defined and undefined.
Reminder
Continuous function graphs do not have breaks within their domain.
Points of discontinuity are places where the graph jumps up or down.
Future topics will cover removable and infinite discontinuities.
Question to Ponder
Is the function continuous at x = -1? Share reasons in comments.