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Intermediate Value Theorem for Continuous Functions

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Function that is Continuous at Irrational Points Only. Bolzano's Theorem ... Continuous Functions (5) Lemma. Corollary. Mika Sepp l : Intermediate Value Theorem ... – PowerPoint PPT presentation

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Title: Intermediate Value Theorem for Continuous Functions


1
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

2
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
3
Continuous Functions (2)
Definition 2
Lemma
Proof
4
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

5
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.

6
Continuous Functions (5)
Lemma
Corollary
7
Continuous Functions (6)
Function which is continuous at irrational points
and discontinuous at rational points.
Example
Define
8
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.
9
Bolzanos Theorem (1)
Bolzanos Theorem
Proof
10
Bolzanos Theorem (2)
BolzanosTheorem
Proof (contd)
11
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.
12
Using the Intermediate Value Theorem (1)
Problem
Solution
13
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.
14
Using the Intermediate Value Theorem (3)
Problem
GraphicalSolution
17th iteration, ??0.45018750
1st iteration, ??0.5
2nd iteration, ??0.25
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