Intermediate Value Theorem for Continuous Functions

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Intermediate Value Theorem for Continuous
Functions

• Review of Properties of Continuous
  FunctionsFunction that is Continuous at
  Irrational Points OnlyBolzanos
  TheoremIntermediate Value Theorem for Continuous
  FunctionsApplications of the Intermediate Value
  Theorem

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Continuous Functions (1)
Definition 1
A function which is not continuous (at a point
or in an interval) is said to be discontinuous.
Continuous function
Discontinuous function
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Continuous Functions (2)
Definition 2
Lemma
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Continuous Functions (2)
Definition 2
Lemma
Proof
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Continuous Functions (3)
The following functions are continuous at all
points where they take finite values.

• Polynomials they are continuous everywhere
• Rational functions
• Functions defined by algebraic expressions
• Exponential functions and their inverses
• Trigonometric functions and their inverses

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Continuous Functions (4)
Assume that f and g are continuous functions.
Theorem

• The following functions are continuous
• f g
• f g
• f / g provided that g ? 0, i.e. the function is
  continuous at all points x for which g(x) ? 0.

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Continuous Functions (5)
Lemma
Corollary
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Continuous Functions (6)
Function which is continuous at irrational points
and discontinuous at rational points.
Example
Define
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Continuous Functions (7)
Define
Continuity at irrational points follows from the
fact that when approximating an irrational number
by a rational number m/n, the denominator n
grows arbitrarily large as the approximation gets
better.
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Bolzanos Theorem (1)
Bolzanos Theorem
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Bolzanos Theorem (1)
Bolzanos Theorem
Proof
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Bolzanos Theorem (1)
Bolzanos Theorem
Proof
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Bolzanos Theorem (2)
BolzanosTheorem
Proof (contd)
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Bolzanos Theorem (2)
BolzanosTheorem
Proof (contd)
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Bolzanos Theorem (2)
BolzanosTheorem
Proof (contd)
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Bolzanos Theorem (2)
BolzanosTheorem
Proof (contd)
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Intermediate Value Theorem for Continuous
Functions
Theorem
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Intermediate Value Theorem for Continuous
Functions
Theorem
Proof
If c gt f(a), apply the previously shown
Bolzanos Theorem to the function f(x) - c.
Otherwise use the function c f(x).
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Intermediate Value Theorem for Continuous
Functions
Theorem
Proof
If c gt f(a), apply the previously shown
Bolzanos Theorem to the function f(x) - c.
Otherwise use the function c f(x).
The Intermediate Value Theorem means that a
function, continuous on an interval, takes any
value between any two values that it takes on
that interval. A continuous function cannot grow
from being negative to positive without taking
the value 0.
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Using the Intermediate Value Theorem (1)
Problem
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Using the Intermediate Value Theorem (1)
Problem
Solution
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Using the Intermediate Value Theorem (1)
Problem
Solution
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Using the Intermediate Value Theorem (2)
Problem
Solution (contd)
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Using the Intermediate Value Theorem (2)
Problem
Solution (contd)
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Using the Intermediate Value Theorem (2)
Problem
Solution (contd)
Repeat the above to find an interval with length
lt0.002 containing the solution. The mid-point of
this interval is the desired approximation.
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Using the Intermediate Value Theorem (3)
Problem
GraphicalSolution
1st iteration, ??0.5
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Using the Intermediate Value Theorem (3)
Problem
GraphicalSolution
1st iteration, ??0.5
2nd iteration, ??0.25
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Using the Intermediate Value Theorem (3)
Problem
GraphicalSolution
17th iteration, ??0.45018750
1st iteration, ??0.5
2nd iteration, ??0.25