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Continuity and exception (singularity)

In order to investigate continuity at singularity point, we consider a function definition which is obtained by modifying modulus function :

| x; x>0 f(x) = || -x: x<0

Graph of function

Graph of function

The function is not defined for x=0. Singularity is a point where function is not defined. Thus, this point is singularity for the function which is otherwise defined for all other points on R. They are equal and are equal to a value which is not defined! In accordance with the definition, the limit of function exists at x=0 and is equal to zero. At this point, both left and right limits exist. After all, limit points to a value. Here, it points to a value outside the domain of function. See graph. Existence of limit at x=0, however, has nothing to do with continuity of function at that point as function is not defined at x=0 in the first place. This point is not the part of function definition i.e. its domain and hence continuity or discontinuity is not a concern.

We consider another function. This is modulus of reciprocal condition, f(x) = |1/x|. Using transformation technique, we draw of graph of function as shown :

Graph of function

Graph of function

The left and right limits both are positive infinity (read tending to infinity) at x=0. The limit of the function is infinity at this point. But, again point x=0 is not part of function definition. Hence, we say that function is continuous in its domain R-{0}.

Clearly, we need to distinguish between “discontinuous” and “undefined”. Going by two illustrations above, we need to understand that tangent, cotangent, secant and cosecant functions are continuous functions though they appear to be discontinuous graphically. They are not defined at certain values, but then these points are not the part of domain as well. As a matter of fact, rational functions, known to have singularities corresponding to points where denominator is zero, are continuous functions in their domain. For this reason, function such as reciprocal function "1/x" is a continuous function in its domain, which is R – {0}.

Continuity and differentiability

Continuity at a point does not guarantee that function is differentiable at the point. In order to understand this, we now consider the modulus function itself. Is function continuous at x=0? Is function is differentiable at x=0?

Graph of function

Graph of function

The limit of function is 0, which is finite and is equal to function value. Clearly, function is continuous at this point – thought we can not draw a tangent at this point and as such function is not differentiable at the point. The converse of the assertion, however, is true. If a function is differentiable at a point, then function is continuous at that point. Clearly, differentiability has stricter requirements than that of continuity.

Types of discontinuity

A function is discontinuous if it is not continuous. We can fail the conditions of continuity in many ways. Accordingly, there are following types of discontinuity :

Questions & Answers

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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
for teaching engĺish at school how nano technology help us
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
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Berger describes sociologists as concerned with
Mueller Reply
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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