<< Chapter < Page Chapter >> Page >
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

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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Analysis of functions of a single variable' conversation and receive update notifications?