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AP Calculus AB - McDermott
Chapter 2
Section 2.3.1 – Continuity
Definition of Continuity
Continuity at a Point: A function is continuous at c if the following three conditions are met.
1. f (c) is defined.
2.
exists.
3.
Continuity on an Open Interval: A function is continuous on an open interval (a, b) if it is
continuous at each point in the interval. A function that is continuous on the entire real line
(-¥,¥) is everywhere continuous.
If a function is not continuous on an open interval is it said to have a discontinuity at c.
Discontinuities fall into two categories: removable and nonremovable.
Example 1 Continuity of a Function
Discuss the continuity of each function.
1
a. f (x) =
x
x 2 -1
b. g(x) =
x -1
ì x + 1, x £ 0
c. h(x) = í 2
î x + 1, x > 0
d. y = sin x
a. The domain of f is all nonzero numbers. Nonremovable discontinuity at x = 0.
b. The domain is all real numbers except 1. Removable discontinuity at x = 1.
c. The domain is all real numbers. It is continuous on the whole number line.
d. The domain is all real numbers and it is continuous on the whole number line.
AP Calculus AB - McDermott
Chapter 2
Section 2.3.2 – Intermediate Value Theorem
The existence of a limit
Let f be a function and let c and L be real numbers. The limit of f (x) as x approaches c is L if
and only if
and
Definition of Continuity on a Closed Interval
A function f is continuous on the closed interval [a, b] if it is continuous on the open interval
(a, b) and
and
.
The function f is continuous from the right at a and continuous from the left at b.
Intermediate Value Theorem – If f is continuous on the closed interval [a, b] and k is any number
between f (a) and f (b), then there is at least one number c in [a, b] such that
f (c) = k.
Use the Intermediate Value Theorem to show that the polynomial function f (x) = x 3 + 2x -1 has a
zero in the interval [0, 1].
First Note that f is continuous on the closed interval [0, 1].
Then plug in the endpoints into the function.
Since the function goes from negative to positive, ZERO has to exist in that interval.
Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1, 5],
there is one with a volume of 275 cubic centimeters.
First note that the volume of a sphere is continuous on the closed interval [1, 5].
Then plug in the endpoints into the function.
pg. 84 ~ 1-9(O), 11-16, 19, 21, 23, 41-50