the
Real Analysis appendices of
Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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These Real Analysis appendices
continue the decimal
viewpoint of limits, continuity and convergence in chapter 14. and
this further lesson
A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits, Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II
Proofs of one-sided theorems could be of interest in the
study of 2D topology.
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Extreme Value Theorems
The previous proof has a consequence not stated. In the
proof, let e = 1. Then there
exists x1,¼,xn
in [a,b] such that if x is in [a,b]
then for some j, |x-xj|
£ d
and |f(x)-f(xj)|
£ e =
1. The last inequality is equivalent to
The latter implies
| -1+f(xj)
£ f(x)
£ 1+f(xj) |
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Put K = -1+min1 £
q £ n f(xq)
and M = +1+max1 £ q
£ nf(xq).
Then K £ f(x)
£ M if x is in the
interval [a,b]. This holds since there exist a
whole number j ³ 1 such
that
| K £
-1+f(xj)
£ f(x)
£ 1+f(xj)
£ M |
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Thus the range W of the function f(x)
is contained in the finite interval [K,M].
This implies the first conclusion in the following
assertion.
Theorem F.6 [Extreme Value Theorem] Suppose b
> a. Suppose f(x) is continuous at
each point in the interval [a,b]. Then there
is a finite interval [K,M] such that f(x)
belongs this interval whenever x belongs to the
domain [a,b]. Further, there exist a unique
pair of real numbers ymin and ymax
such ymin £
f(x) £ ymax
with the following properties.
- There is at least one real number v in the
interval [a,b] such that f(v)
= ymin; and
- there is at least one real number w in the
interval [a,b] such that f(w)
= ymax.
Proof of Extreme Value Theorem.
As shown above, there is an finite interval [K,M]
which contains all the points in the range of f,
that is, the set
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| {y
| there exists a point x in
[a,b] with y = f(x)} |
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of the function f. Since W is contained in a
finite interval [K,M], a previous theorem
implies there is a smallest interval [A,B] Ì
[K,M] which contains W.
Let ymin = A = inf(W).
Now ymin is either in the range W,
or it is a limit point of the range W. In the first
case, there is nothing to do. In the second, the closed
range theorem implies there is at least one real number v
in the interval [a,b] such that f(v)
= ymin.
Similarly, let ymax = B
= sup(W). Then ymax is
either in the range W, or it is a limit point of
the range W. In the first case, there is nothing to
do. In the second, the closed range theorem implies there
is at least one real number v in the interval [a,b]
such that f(v) = ymax.
Note the intermediate value theorem implies the range of f
equals the interval [ymin,ymax].
The numbers ymin = absolute
minimum and ymax = absolute
maximum are called the extreme values of the function f(x)
on the interval [a,b]. A function f(x)
may have different extreme values on different intervals in
its domain.
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