# 5.12 Continuous function

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Idea of "continuity" in the context of function is same as its dictionary meaning. It simply means that function is continuous without any abrupt or sudden change in the value of function. An indicative way to test continuity is to see (graphically or otherwise) that there is no sudden jump in function value in the domain of function. If domain of function is a continuous interval i.e. no points or sub-intervals are excluded from real number set representing domain, then we can draw a continuous function without lifting the drawing instrument i.e. pen, pencil etc. If we have to lift the pen in order to complete the graph of a function in the continuous interval, then function is not continuous.

The feature of continuity is related to function. Therefore, continuity of a function is meaningful in its domain only. It means that we do not need to evaluate continuity in intervals or points where function is not defined. This is an important consideration that helps us to differentiate between “discontinuity” and “undefined values”.

## Important concepts

The condition of continuity is expressed in terms of limit and function value. Both of these are required to exist and be equal. We shall learn about these aspects more in detail after having brief overview of these terms.

## Limit from left

The limit from left means that a function approaches a value ( ${L}_{l}$ ) as x approaches the test point “a” from left such that x is always less than “a” – but not equal to “a”.

$\underset{x>a-}{\overset{}{\mathrm{lim}}}f\left(x\right)={L}_{l}$

We show here three illustrations of “limit from left”. The important aspect of these figures is that graph tends to a particular value (infinity is also included). This is done by showing the orientation of graph pointing to limiting value when x is infinitesimally close to test point. Important point to note is that graph does not reach limiting value. Note empty circle at the end of graph, which represents the value of limit not yet occupied by graph. Similarly, asymptotic nature of graph tending to infinity maintains a small distance away from asymptotes, denoting that graph does not reach limiting value.

## Limit from right

The limit from right means that a function approaches a value ( ${L}_{r}$ ) as x approaches the test point “a” from right such that x is always greater than “a” – but not equal to “a”.

$\underset{x>a+}{\overset{}{\mathrm{lim}}}f\left(x\right)={L}_{r}$

We show here three illustrations of “limit from right" as in the earlier case. Important point to note is that graph does not reach limiting value, which represents the value of limit not yet occupied by graph.

## Limit at a point

The limit at a point means that a function approaches a value (L) as x approaches the test point “a” from either side.

$\underset{x>a}{\overset{}{\mathrm{lim}}}f\left(x\right)={L}_{l}={L}_{r}=L$

We show here three illustrations of limit at a point. Important point to again note is that graph does not reach limiting value, which represents the value of limit not yet occupied by graph.

## Function value

Function value is obtained by substituting x values in the function. In case of rational function, we first reduce expression by removing common factors from numerator and denominator.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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