Section 2.4 Continuity Math 1a October 3, 2007
Jun 27, 2015
Section 2.4Continuity
Math 1a
October 3, 2007
Questions
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseAt one point in your life you were exactly three feet tall.
Questions
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseAt one point in your life you were exactly three feet tall.
Questions
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseAt one point in your life you were exactly three feet tall.
Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain off , then
limx→a
f (x) = f (a)
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Definition of Continuity
DefinitionLet f be a function defined near a. We say that f is continuous ata if
limx→a
f (x) = f (a).
Free Theorems
Theorem
(a) Any polynomial is continuous everywhere; that is, it iscontinuous on R = (−∞,∞).
(b) Any rational function is continuous wherever it is defined; thatis, it is continuous on its domain.
Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on its whole domain, which is[−1
4 ,∞).
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Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on its whole domain, which is[−1
4 ,∞).
Showing a function is continuous
Example
Let f (x) =√
4x + 1. Show that f is continuous at 2.
SolutionWe have
limx→a
f (x) = limx→2
√4x + 1
=√
limx→2
(4x + 1)
=√
9 = 3.
Each step comes from the limit laws.
In fact, f is continuous on its whole domain, which is[−1
4 ,∞).
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The Limit Laws give Continuity Laws
TheoremIf f and g are continuous at a and c is a constant, then thefollowing functions are also continuous at a:
1. f + g
2. f − g
3. cf
4. fg
5.f
g(if g(a) 6= 0)
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Transcendental functions are continuous, too
TheoremThe following functions are continuous wherever they are defined:
1. sin, cos, tan, cot sec, csc
2. x 7→ ax , loga, ln
3. sin−1, tan−1, sec−1
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What could go wrong?
In what ways could a function f fail to be continuous at a point a?Look again at the definition:
limx→a
f (x) = f (a)
Pitfall #1
: The limit does not exist
Example
Let
f (x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
At which points is f continuous?
SolutionAt any point a in [0, 2] besides 1, lim
x→af (x) = f (a) because f is
represented by a polynomial near a, and polynomials have thedirect substitution property. However,
limx→1−
f (x) = limx→1−
x2 = 12 = 1
limx→1+
f (x) = limx→1+
2x = 2(1) = 2
So f has no limit at 1. Therefore f is not continuous at 1.
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Pitfall #1: The limit does not exist
Example
Let
f (x) =
{x2 if 0 ≤ x ≤ 1
2x if 1 < x ≤ 2
At which points is f continuous?
SolutionAt any point a in [0, 2] besides 1, lim
x→af (x) = f (a) because f is
represented by a polynomial near a, and polynomials have thedirect substitution property. However,
limx→1−
f (x) = limx→1−
x2 = 12 = 1
limx→1+
f (x) = limx→1+
2x = 2(1) = 2
So f has no limit at 1. Therefore f is not continuous at 1.
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Pitfall #2
: The function has no value
Example
Let
f (x) =x2 + 2x + 1
x + 1
At which points is f continuous?
SolutionBecause f is rational, it is continuous on its whole domain. Notethat −1 is not in the domain of f , so f is not continuous there.
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Pitfall #2: The function has no value
Example
Let
f (x) =x2 + 2x + 1
x + 1
At which points is f continuous?
SolutionBecause f is rational, it is continuous on its whole domain. Notethat −1 is not in the domain of f , so f is not continuous there.
Pitfall #3
: function value 6= limit
Example
Let
f (x) =
{46 if x 6= 1
π if x = 1
At which points is f continuous?
Solutionf is not continuous at 1 because f (1) = π but lim
x→1f (x) = 46.
Pitfall #3: function value 6= limit
Example
Let
f (x) =
{46 if x 6= 1
π if x = 1
At which points is f continuous?
Solutionf is not continuous at 1 because f (1) = π but lim
x→1f (x) = 46.
Special types of discontinuites
removable discontinuity The limit limx→a
f (x) exists, but f is not
defined at a or its value at a is not equal to the limitat a.
jump discontinuity The limits limx→a−
f (x) and limx→a+
f (x) exist, but
are different. f (a) is one of these limits.
The greatest integer function f (x) = [[x ]] has jump discontinuities.
Special types of discontinuites
removable discontinuity The limit limx→a
f (x) exists, but f is not
defined at a or its value at a is not equal to the limitat a.
jump discontinuity The limits limx→a−
f (x) and limx→a+
f (x) exist, but
are different. f (a) is one of these limits.
The greatest integer function f (x) = [[x ]] has jump discontinuities.
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A Big Time Theorem
Theorem (The Intermediate Value Theorem)
Suppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
Illustrating the IVT
Suppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b]
and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b]
and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b).
Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
c
c1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
cc1 c2 c3
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
c
c1 c2 c3
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2]. Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2].
Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2]. Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.
Using the IVT
Example
Prove that the square root of two exists.
Proof.Let f (x) = x2, a continuous function on [1, 2]. Note f (1) = 1 andf (2) = 4. Since 2 is between 1 and 4, there exists a point c in(1, 2) such that
f (c) = c2 = 2.
In fact, we can “narrow in” on the square root of 2 by the methodof bisections.