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Starter Guide (Views)
Real Player Videos
2. Limit of a Sequence
Triangle Inequality
One Side Range Theorem
Range On One Side
Starter & Warm Up Lessons
1. Usual Review/Starter Lessons
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material
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Range and Intermediate Value Theorems
In the previous lesson, we proved the following.
One Sided Range Theorem: Let I be a non-empty interval. Suppose f: I
--> R is an equi-continuous function or Lipshitz Continuous function on I.
Suppose M is real number not in the range f(I) of f and not a limit
point of f(I). Then either (i) for all x in I, f(x) < M or (ii)
for all x in I, f(x) > M.
Contrapositive Version: If M is a real number for which there
are points a and b with f(a) < M < f(b) then M is a limit
point of f(I) or M is in the image f(I).
Intermediate Value Theorem: Let I be a finite interval [a,b]. Suppose
f: I --> R is an equi-continuous function or Lipshitz Continuous function on
[a,b]. If a real number M satisfies f(c) < M <
f(d) for some points c and d in I then M belongs to the range of f and M =
f(q) for some point q in the interval [a,b]..
Proof: The Contrapositive version of the one side range theorem
implies M is the image f([a,b]), in which case there is nothing to more to
show, or M is a limit point of the image f([a,b]). But the image of a closed
interval [a,b] under a continuous function is closed. That is, it contains all
its limit points. So again M is in the image f([a,b]).
If I is a finite interval [a, b] then continuity
of f(x) on I implies equicontinuity on I. That implies the following.
Usual Intermediate Value Theorem: Suppose f: [a, b] -->
R is an continuous. Suppose for some numbers a < b in I, the real
number y belongs to the interval with end points f(a) and f(b). Then there
a least one point x in the interval with endpoints a and b such that f(c) =
y.
Extreme Value Theorem Revisited. Suppose b > a.
Suppose f(x) is continuous at each point in the interval [a,b].
Further, there exist a unique pair of real numbers ymin
and ymax such that f([a,b]) = [ymin,
ymax].
In other words, the image of a closed interval [a, b] under a continuous,
equi-continous or Lipshitz continuous functions is a closed interval.
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