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Finally, using the continuity of both f and g applied to the positive numbers ϵ 1 = ϵ / ( 4 M 2 | g ( c ) | 2 ) and ϵ 2 = ϵ / ( 4 M 1 | g ( c ) | 2 ) , choose δ > 0 , with δ < min ( δ 1 , δ 2 , δ ' ) , and such that if | y - c | < δ and y S then | f ( y ) - f ( c ) | < ϵ 4 M 2 / | g ( c ) | 2 and | g ( c ) - g ( y ) | < ϵ 4 M 1 / | g ( c ) | 2 . Then, if | y - c | < δ and y S we have that

| f ( y ) g ( y ) - f ( c ) g ( c ) | < ϵ

as desired.

  1. Prove part (2) of the preceding theorem. (It's an ϵ / 2 argument.)
  2. Prove part (3) of the preceding theorem. (It's similar to the proof of part (5) only easier.)
  3. Prove part (4) of the preceding theorem.
  4. Prove part (6) of the preceding theorem.
  5. Suppose S is a subset of R . Verify the above theoremreplacing “ continuity” with left continuity and right continuity.
  6. If S is a subset of R , show that f is continuous at a point c S if and only if it is both right continuous and left continuous at c .

The composition of continuous functions is continuous.

Let S , T , and U be subsets of C , and let f : S T and g : T U be functions. Suppose f is continuous at a point c S and that g is continuous at the point f ( c ) T . Then the composition g f is continuous at c .

Let ϵ > 0 be given. Because g is continuous at the point f ( c ) , there exists an α > 0 such that | g ( t ) - g ( f ( c ) ) | < ϵ if | t - f ( c ) | < α . Now, using this positive number α , and using the fact that f is continuous at the point c , there exists a δ > 0 so that | f ( s ) - f ( c ) | < α if | s - c | < δ . Therefore, if | s - c | < δ , then | f ( s ) - f ( c ) | < α , and hence | g ( f ( s ) ) - g ( f ( c ) ) | = | g f ( s ) - g f ( c ) | < ϵ , which completes the proof.

  1. If f : C C is the function defined by f ( z ) = z , prove that f is continuous at each point of C .
  2. Use part (a) and Theorem 3.2 to conclude that every rational function is continuous on its domain.
  3. Prove that a step function h : [ a , b ] C is continuous everywhere on [ a , b ] except possibly at the points of the partition P that determines h .
  1. Let S be the set of nonnegative real numbers, and define f : S S by f ( x ) = x . Prove that f is continuous at each point of S . HINT: For c = 0 , use δ = ϵ 2 . For c 0 , use the identity
    y - c = ( y - c ) y + c y + c = y - c y + c y - c c .
  2. If f : C R is the function defined by f ( z ) = | z | , show that f is continuous at every point of its domain.

Using the previous theorems and exercises, explain why the following functions f are continuous on their domains. Describe the domains as well.

  1. f ( z ) = ( 1 - z 2 ) / ( 1 + z 2 ) .
  2. f ( z ) = | 1 + z + z 2 + z 3 - ( 1 / z ) | .
  3. f ( z ) = 1 + 1 - | z | 2 .
  1. If c and d are real numbers, show that max ( c , d ) = ( c + d ) / 2 + | c - d | / 2 .
  2. If f and g are functions from S into R , show that max ( f , g ) = ( f + g ) / 2 + | f - g | / 2 .
  3. If f and g are real-valued functions that are both continuous at a point c , show that max ( f , g ) and min ( f , g ) are both continuous at c .

Let N be the set of natural numbers, let P be the set of positive real numbers, and define f : N P by f ( n ) = 1 + n . Prove that f is continuous at each point of N . Show in fact that every function f : N C is continuous on this domain N .

HINT: Show that for any ϵ > 0 , the choice of δ = 1 will work.

  1. Negate the statement: “For every ϵ > 0 , | x | < ϵ . ' '
  2. Negate the statement: “For every ϵ > 0 , there exists an x for which | x | < ϵ . ' '
  3. Negate the statement that “ f is continuous at c . ' '

The next result establishes an equivalence between the basic ϵ , δ definition of continuity and a sequential formulation.In many cases, maybe most, this sequential version of continuity is easier to work with than the ϵ , δ version.

Let f : S C be a complex-valued function on S , and let c be a point in S . Then f is continuous at c if and only if the following condition holds: For every sequence { x n } of elements of S that converges to c , the sequence { f ( x n ) } converges to f ( c ) . Or, said a different way, if { x n } converges to c , then { f ( x n ) } converges to f ( c ) . And, said yet a third (somewhat less precise) way, the function f converts convergent sequences to convergent sequences.

Suppose first that f is continuous at c , and let { x n } be a sequence of elements of S that converges to c . Let ϵ > 0 be given. We must find a natural number N such that if n N then | f ( x n ) - f ( c ) | < ϵ . First, choose δ > 0 so that | f ( y ) - f ( c ) | < ϵ whenever y S and | y - c | < δ . Now, choose N so that | x n - c | < δ whenever n N . Then if n N , we have that | x n - c | < δ , whence | f ( x n ) - f ( c ) | < ϵ . This shows that the sequence { f ( x n ) } converges to f ( c ) , as desired.

We prove the converse by proving the contrapositive statement; i.e., we will show that if f is not continuous at c , then there does exist a sequence { x n } that converges to c but for which the sequence { f ( x n ) } does not converge to f ( c ) . Thus, suppose f is not continuous at c . Then there exists an ϵ 0 > 0 such that for every δ > 0 there is a y S such that | y - c | < δ but | f ( y ) - f ( c ) | ϵ 0 . To obtain a sequence, we apply this statement to δ 's of the form δ = 1 / n . Hence, for every natural number n there exists a point x n S such that | x n - c | < 1 / n but | f ( x n ) - f ( c ) | ϵ 0 . Clearly, the sequence { x n } converges to c since | x n - c | < 1 / n . On the other hand, the sequence { f ( x n ) } cannot be converging to f ( c ) , because | f ( x n ) - f ( c ) | is always ϵ 0 .

This completes the proof of the theorem.

Questions & Answers

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
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
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
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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