Closed Range Theorem

A set W is set to be closed if and only if it contains all its limit points. Recall, a real number A to be a limit point of a infinite set of numbers S if and only if every interval I centered at A, no matter how small, contains infinitely many elements of the set S. Equivalently, a real number A to be a limit point of a set S if and only if every interval I centered at A contains at least one element s ¹ A of the set S.

Theorem F.2 [Closed Range Theorem] Suppose f(x) is a real-valued function which is continuous at each point x in the interval [a,b]. Suppose if L is a limit point of the function's range

range(f) = {y | there exists a point x in [a,b] such that y = f(x)}.

Then there exists a point w in [a,b] such that L = f(w). The proof relies on the second triangle inequality

|a-b| ³

ê ê ê

|a|-|b|

ê ê ê

Proof of Closed Range Theorem.

Observe L is a limit point of the range of the function f, that is, the set

W = {y |  there exists a point x in [a,b] such that y = f(x)}

Therefore, each interval of length [1/2]10^-k centered at L contains at least one point of W. Choose one and label it w_k. Then w_k = f(x_k) for some x_k in the interval [a,b]. Further, the infinite set of points x_k has a greatest lower bound A = inf{x|  k ³ 1} in the interval [a,b].

For the sake of a contradiction, suppose that f(A) ¹ L. Then there exists an integer K > 0 such that |f(A)-L| > [1/2] [1/(10^K)] for some whole number K. Now in every interval centered at A, there exists a w_k with k > K. This w_k has the property that

 |f(w_k)-L| £ [1/2]10^-k < [1/2]10^-K.

Therefore

Presenting Appendices from  Volume 3, Why Slopes and More Math,  If the  epsilon-delta viewpoint of limits, continuity and convergence is not yet comfortable, see  Chapters 14 to 19 in Volume 3 are related.  

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.

Vol 1A Logic Postscripts
online only include

Proof by Absurdity alias proof by contradiction

How the demand for consistency supports the law of the excluded middle

Reality versus or with the aid of Imagination

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