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Function Limits VI: Continuous Functions
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Lesson Summary:
We define both an intuition and a rigorous mathematical definition of continuous functions.
Lesson Inputs:
Function Limits V: Properties of Function Limits
Function Limits III: The Gory Details
Lesson Outputs:
Antiderivatives
Geometry of Functions II: The Extreme-Value Theorem
Differentiation III: How Differentiability and Continuity Are Related
A List of Continuous Functions
The Intermediate-Value Theorem
Properties of Continuous Functions
Function Limits VII: Putting It All Together With Useful Examples
GAIN AN ADVANTAGE
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Easy to digest lessons
Cheat sheets
PDF: How to Make an A+ in Your First Calculus Course
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Lesson Specific Problems
If
f
(
x
)
is continuous and
m
(
x
)
is the minimum value of
f
(
x
)
in the closed interval
[
x
,
a
]
, then prove that
lim
If
f(x)
is continuous and
M(x)
is the maximum value of
f(x)
in the closed interval
[a,x]
, then prove that
\lim_{x\to a^+}\,M(x)=f(a).
If
f(x)
is continuous and
m(x)
is the minimum value of
f(x)
in the closed interval
[a,x]
, then prove that
\lim_{x\to a^+}\,m(x)=f(a).
Prove that if
f(x)
is continuous and
K
is a constant, then
f(x)+K
is a continuous function.
Prove that if
f(x)
is continuous on
[a,b]
and
f(a)\lt 0\lt f(b)
or
f(b)\lt 0\lt f(a)
, then there exists a number
K
between
a
and
b
such that
f(K)=0
.
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