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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Appendix G
Differentiable Functions
Rolle's Theorem
A real-valued function f(x) is said to be
differentiable at a point x = a if and only if
| f¢(x)
= |
lim
Dx->
0
|
|
f(x+Dx)-f(x)
D |
|
|
exists, that is, the chord slope [(f(x+Dx)-f(x))/(D)]
approaches a finite limit as Dx->
0. As noted in the earlier discussion of the linear
approximation y = f(x1)+f¢(x1)(x-x1)
» f(x), if a
function is differentiable at a point x1
then it continuous at x1, meaning limx->
x1 f(x) exists and
equals the real number f(x1).
Theorem G.1 [Rolle's Theorem] Suppose f(x)
is continuous on an interval [a,b] and
differentiable when a < x < b.
Further suppose f(a) = f(b).
Then there exists a point c interior to the interval
[a,b] with f¢(c)
= 0. Note: saying the point c interior to the
interval [a,b] means a < c
< b.
Proof of Rolle's Theorem.
In general, the extreme value theorem for continuous
functions implies there exist at least two points xmin
and xmax in the interval [a,b]
with the property that f(xmin)
£ f(x) £
f(xmax) for every x
in the interval [a,b]. In particular, this
property implies at the end points that f(xmin)
£ f(a) = f(b)
£ f(xmax).
If f(xmin) < f(a)
= f(b) then a < xmin
< b since xmin in [a,b]
cannot be an end-point. But then f¢(xmin)
= 0 since otherwise, an interior minimum would not occur
at xmin. (See the earlier
discussion of linear approximation.) Thus the conclusion
holds with c = xmin.
Now if f(xmin) = f(a)
= f(b) then either f(xmax)
> f(a) = f(b) or f(xmax)
= f(a) = f(b). In the first
case f(xmax) > f(a)
implies a < xmax < b
and hence f¢(xmax)
= 0, else xmax would not yield an
interior maximum of f(x). In this first
case, the conclusion holds with c = xmin.
Now in the remaining second case f(xmax)
= f(b) = f(xmin).
In this case, the function f(x) is constant
on the interval [a,b]. Thus f¢(c)
= 0 for every point c in the interval. The
conclusion is obvious.
Note that the function f(x) in the above
discussion could have several minima and maxima interior to
the interval. At these points the slope or derivative f¢(c)
= 0.
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