1.5 Continuity
In real life we think of things with continuity as not having interruptions or abrupt changes. This is closely related to the concept of continuity in mathematics. Determine where the function below is not continuous and why.
Jan 22­11:35 AM
1
Determine where each of the following functions is discontinuous.
(see p. 40 ex.8)
Each graph below is discontinuous at one or more points. In examples a and c, the discontinuities are said to be "removable" because we could redefine the function at one point and make the function continuous. In example b, we have an infinite discontinuity, and in example d, we have jump discontinuities. Jan 23­12:36 PM
2
Explain why the graph is discontinuous at the given number a. Then sketch the graph of the function.
A function f is continuous from the right at a number a if A function f is continuous from the left at a number a if Determine at which points of discontinuity each function below is continuous from the left, from the right, or neither.
Jan 23­1:35 PM
3
Graph the piecewise function below and find the numbers at which the function is discontinuous. At which of these points is the function continuous from the left, from the right, or neither?
A function f is continuous on an interval if it is continuous at every number in the interval.
If f and g are continuous at a and c is a constant, then the following functions are also continuous at a.
f + g f ­ g cf fg f/g (if g(a)≠0)
The following types of functions are continuous at every number in their domains: polynomial functions, rational functions, root functions, and trigonometric functions. On what intervals is each of the following functions continuous?
Jan 23­12:42 PM
4
Another way of combining functions to get a new continuous function is to form the composite function fog. This fact is a consequence of the following theorem.
The limit symbol can be moved
through a function symbol if the
function is continuous and the limit
exists.
A continuous function
of a continuous function
is continuous.
Explain why each function below is continuous at every number in its domain. State the domain.
Jan 23­1:14 PM
5
Specifically, the Intermediate Value Theorem can be used to locate roots or zeros of a continuous function between two values a and b by showing that the signs of f(a) and f(b) are different, thus proving that f(c) = 0 for some value of c between a and b.
Given the function
use the Intermediate Value Theorem to locate a zero between two integers. Jan 23­1:33 PM
6