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We collect here some theorems that show some of the consequences of continuity.Some of the theorems apply to functions either of a real variable or of a complex variable,while others apply only to functions of a real variable. We begin with what may be the most famous such result, and this one is about functions of a real variable.

We collect here some theorems that show some of the consequences of continuity.Some of the theorems apply to functions either of a real variable or of a complex variable,while others apply only to functions of a real variable. We begin with what may be the most famous such result, and this one is about functions of a real variable.

Intermediate value theorem

If f : [ a , b ] R is a real-valued function that is continuous at each point of the closed interval [ a , b ] , and if v is a number (value) between the numbers f ( a ) and f ( b ) , then there exists a point c between a and b such that f ( c ) = v .

If v = f ( a ) or f ( b ) , we are done. Suppose then, without loss of generality, that f ( a ) < v < f ( b ) . Let S be the set of all x [ a , b ] such that f ( x ) v , and note that S is nonempty and bounded above. ( a S , and b is an upper bound for S . ) Let c = sup S . Then there exists a sequence { x n } of elements of S that converges to c . (See [link] .) So, f ( c ) = lim f ( x n ) by [link] . Hence, f ( c ) v . (Why?)

Now, arguing by contradiction, if f ( c ) < v , let ϵ be the positive number v - f ( c ) . Because f is continuous at c , there must exist a δ > 0 such that | f ( y ) - f ( c ) | < ϵ whenever | y - c | < δ and y [ a , b ] . Since any smaller δ satisfies the same condition, we may also assume that δ < b - c . Consider y = c + δ / 2 . Then y [ a , b ] , | y - c | < δ , and so | f ( y ) - f ( c ) | < ϵ . Hence f ( y ) < f ( c ) + ϵ = v , which implies that y S . But, since c = sup S , c must satisfy c y = c + δ / 2 . This is a contradiction, so f ( c ) = v , and the theorem is proved.

The Intermediate Value Theorem tells us something qualitative about the range of a continuous function on an interval [ a , b ] . It tells us that the range is “connected;” i.e., if the range contains two points c and d , then the range contains all the points between c and d . It is difficult to think what the analogous assertion would be for functions of a complex variable, since “between” doesn't mean anything for complex numbers.We will eventually prove something called the Open Mapping Theorem in [link] that could be regarded as the complex analog of the Intermediate Value Theorem.

The next theorem is about functions of either a real or a complex variable.

Let f : S C be a continuous function, and let C be a compact (closed and bounded) subset of S . Then the image f ( C ) of C is also compact. That is, the continuous image of a compact set is compact.

First, suppose f ( C ) is not bounded. Thus, let { x n } be a sequence of elements of C such that, for each n , | f ( x n ) | > n . By the Bolzano-Weierstrass Theorem, the sequence { x n } has a convergent subsequence { x n k } . Let x = lim x n k . Then x C because C is a closed subset of C . Co, f ( x ) = lim f ( x n k ) by [link] . But since | f ( x n k ) | > n k , the sequence { f ( x n k ) } is not bounded, so cannot be convergent. Hence, we have arrived at a contradiction, and the set f ( C ) must be bounded.

Now, we must show that the image f ( C ) is closed. Thus, let y be a limit point of the image f ( C ) of C , and let y = lim y n where each y n f ( C ) . For each n , let x n C satisfy f ( x n ) = y n . Again, using the Bolzano-Weierstrass Theorem, let { x n k } be a convergent subsequence of the bounded sequence { x n } , and write x = lim x n k . Then x C , since C is closed, and from [link]

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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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