# 3.5 Deeper analytic properties of continuous functions

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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 $f:\left[a,b\right]\to R$ 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]

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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Daniel
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Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
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Harper
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s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
types of nano material
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Porter
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Yasmin
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Cesar
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Uday
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preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
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Azam
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Prasenjit
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Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
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Azam
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Uday
I'm interested in Nanotube
Uday
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Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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