1. 2.3 Continuity

2. Definition: A function f is continuous at a number a if
Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil.
Definition: A function f is continuous at a number a if
(the limit is the same as the value of the function). 1 2 3 4
This function is discontinuous (has discontinuities) at x=1 and x=2.
It is continuous everywhere else on the interval [0,4].

3. Removable Discontinuities:
(You can fill the hole.)
Essential Discontinuities:
jump
infinite

4. Definition: A function f is continuous from the right at a number a if
and f is continuous from the left at a number a if
Definition: A function f is continuous on an interval if it is continuous at every number in the interval.
Examples on the board.

5. Theorem: If f and g are continuous at a and c is a constant, then the following functions are also continuous at a:
Theorem: The following types of functions are continuous at every number in their domains: polynomials, rational functions, root functions, trigonometric functions.
Example: The function
is continuous on the intervals [0,2) and (2,)