Calculus 2.3

1
2.3 Continuity
Grand Canyon, Arizona
2
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.
A function is continuous at a point if the limit
is the same as the value of the function.
Where is this function continuous?
3
The analytic definition of Continuity at a point
is
Closed circle
Two sides of a curve leading to the open circle
4
Closed circle
Closed circle
Two sides of a curve leading to the open circle
Two sides of a curve leading to the open circle
and, therefore, f (x) is continuous at x 3
5
Removable Discontinuities
(You can fill the hole.)
Essential Discontinuities
oscillating
infinite
jump
6
Removing a discontinuity
7
Removing a discontinuity
Another way to express the solution could be
8
Removing a discontinuity
9
Continuous functions can be added, subtracted,
multiplied, divided and multiplied by a constant,
and the new function remains continuous.
examples
10
Intermediate Value Theorem
If a function is continuous between a and b, then
it takes on every value between and
.
11
Example 5
Is any real number exactly one less than its cube?
(Note that this doesnt ask what the number is,
only if it exists.)
1 solve
p