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Section 2.3 Continuity AP Calculus
**Any function whose graph can be sketched in one continuous motion without lifting the pencil is an example of a
continuous function**
A function is said to be continuous at c if the following 3 conditions are met:
1) f(c) is defined
2) lim f(x) exists
3) lim f(x) = f(c)
xc
xc
Food for Thought: Can you draw a function that violates only 1 or 2 of the above?
*A function that fails to satisfy one or more of the conditions at x=c is said to be discontinuous at x=c.
*A function is said to be continuous on an interval if and only if it is continuous at each point of the interval.
Example: Let f(x) = x+4 for all real x. Show that f(x) is continuous at x=7
1)
2)
3)
 x  1; x  2
. Is f(x) continuous at x = 2?
 3; x  2
Example: Let f ( x)  
Example:
(1a) Does f(-1) exist?
(1b) Does lim f ( x) exist?
(1c) Does lim f ( x) exist?
(1d) If f(x) continuous at x = -1?
x  1
x  1
(2a) Does f(1) exist?
(2b) Does lim f ( x) exist?
(2c) Does lim f ( x) exist?
(2d) If f(x) continuous at x = 1?
x 1
(3a) Is f(x) defined at x = 2?
x 1
(3b) Is f(x) continuous at x = 2?
(4) At what values of x is f(x) continuous?
(5) What new value should be assigned to f(2) to make the new function continuous at x = 2?
(6) What new value should be assigned to f(1) to make the new function continuous at x = 1?
Section 2.3 Continuity AP Calculus
Two main types of discontinuity
1) Removable Discontinuity (Hole)
2) Non Removable Discontinuity (Infinite (vertical asymptote) and Jump (piecewise function))
Example: Is the function continuous? If the function is not continuous, what type of discontinuity exists
and at what values of x.
a) f ( x) 
x2  x
x 1
b) f ( x) 
1
x 1
Continuous Functions
*A function is continuous on an interval if and only if it is continuous at ever point of the interval (continuous function
is continuous on its domain)
Think about it: Is the reciprocal function a continuous function?
Composites
“All composites of continuous functions are continuous”
If f(x) is continuous at c and g(x) is continuous at f(c), then the composite g(f(x)) is continuous at c.
Example: Show that y 
x cos x
is continuous.
x2  4
Intermediate Value Theorem (IVT): a basic result which states that a continuous function on an interval cannot skip
values.
Theorem: If f(x) is continuous on a closed interval [a, b] and f(a)  f(b), then for every value M between f(a) and f(b),
there exists at least one value c  (a, b) such that f(c) = M.
*Corollary: If f(x) is continuous on [a, b] and if f(a) and f(b) are nonzero and
have opposite signs, then f(x) has a zero in (a, b).
Food for Thought: Consider a plane that takes off and climbs from 0 to 20, 000 feet in twenty minutes. The plane
must reach every altitude between 0 and 20, 000 during this twenty minute interval. Thus, at some point the plane
must have had an altitude of exactly 1231 feet. Of course we are assuming the plane’s motion is continuous and that
it cannot jump abruptly.
Formally Stated: Let A(t) be the plane’s altitude at time t. The IVT then asserts that for every altitude M
between 0 and 20,000 feet, there is a time c between 0 and 20 such that A(c) = M. In other words, the graph
of A(t) must intersect the horizontal line y = M.
Section 2.3 Continuity AP Calculus
Example: Prove that the equation sin x = 0.3 has at least one solution.
Example: Show that f(x) = cos2 x – 2 sin ¼ x has a zero in (0, 2). Then locate the zero more accurately using the
Bissection Method.