3.5 Deeper analytic properties of continuous functions

 Page 1 / 2
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 $f:\left[a,b\right]\to R$ is a real-valued function that is continuous at each point of the closed interval $\left[a,b\right],$ and if $v$ is a number (value) between the numbers $f\left(a\right)$ and $f\left(b\right),$ then there exists a point $c$ between $a$ and $b$ such that $f\left(c\right)=v.$

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

Now, arguing by contradiction, if $f\left(c\right) let $ϵ$ be the positive number $v-f\left(c\right).$ Because $f$ is continuous at $c,$ there must exist a $\delta >0$ such that $|f\left(y\right)-f\left(c\right)|<ϵ$ whenever $|y-c|<\delta$ and $y\in \left[a,b\right].$ Since any smaller $\delta$ satisfies the same condition, we may also assume that $\delta Consider $y=c+\delta /2.$ Then $y\in \left[a,b\right],\phantom{\rule{3.33333pt}{0ex}}|y-c|<\delta ,$ and so $|f\left(y\right)-f\left(c\right)|<ϵ.$ Hence $f\left(y\right) which implies that $y\in S.$ But, since $c=supS,$ $c$ must satisfy $c\ge y=c+\delta /2.$ This is a contradiction, so $f\left(c\right)=v,$ and the theorem is proved.

The Intermediate Value Theorem tells us something qualitative about the range of a continuous function on an interval $\left[a,b\right].$ 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 $f:S\to C$ be a continuous function, and let $C$ be a compact (closed and bounded) subset of $S.$ Then the image $f\left(C\right)$ of $C$ is also compact. That is, the continuous image of a compact set is compact.

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

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

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!