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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
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
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
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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