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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]

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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Maciej
characteristics of micro business
Abigail
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Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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