# 1.2 Continuity and convergence of functions

 Page 1 / 1
Definitions of continuity and convergence for functions defined on metric spaces.

## Continuity for functions

Definition 1 A function $f:\left(X,{d}_{x}\right)\to \left(Y,{d}_{y}\right)$ is continuous at a point ${x}_{0}\in X$ if: for any $ϵ>0$ there exists a $\delta >0$ such that if ${d}_{x}\left({x}_{0},{x}_{1}\right)<\delta$ , then ${d}_{y}\left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right)<ϵ$ .

Definition 2 A function: $f:\left(X,{d}_{x}\right)\to \left(Y,{d}_{y}\right)$ is uniformly continuous if: for every $ϵ>0$ there exists a $\delta >0$ such that for all ${x}_{0}\in X$ : if ${d}_{x}\left({x}_{0},{x}_{1}\right)<\delta$ , then ${d}_{y}\left(f\left({x}_{0}\right),f\left({x}_{1}\right)\right)<ϵ$ .

The difference between these definitions is that for a function to be simply continuous at a point, one need only find a $\delta$ for the given input ${x}_{0}$ , while for a function to be uniformly continuous, one needs to find for a given $ϵ$ a single value of $\delta$ that works in the definition for every point over which the function is defined.

Example 1 Consider the function $f:\left(C\left[T\right],{d}_{\infty }\right)\to \left(\mathbb{R},{d}_{0}\right)$ defined as $f\left(x\right)=x\left({t}_{0}\right)$ . In words, the input to $f$ is a continuous function over $T$ , and the output from $f$ is the value of the input function evaluated at $t={t}_{0}$ . We ask the question: Is $f$ continuous at some function input ${x}_{1}\left(t\right)$ ?

• Designate a pair of functions ${x}_{1}\left(t\right),{x}_{2}\left(t\right)$ such that: ${d}_{\infty }\left({x}_{1}\left(t\right),{x}_{2}\left(t\right)\right)<\delta$ , where
${d}_{\infty }\left({x}_{1}\left(t\right),{x}_{2}\left(t\right)\right)=\underset{t\in T}{sup}|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|.$
In other words, $su{p}_{t\in T}|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|<\delta$ .
• Next, we see that ${d}_{0}\left(f\left({x}_{1}\right),f\left({x}_{2}\right)\right)=|{x}_{1}\left({t}_{0}\right)-{x}_{2}\left({t}_{0}\right)|$ .
• Because $|{x}_{1}\left({t}_{0}\right)-{x}_{2}\left({t}_{0}\right)|\le {sup}_{t\in T}|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|$ , by the definition of the supremum, we can simply select $\delta =ϵ$ to get that if ${d}_{0}\left(f\left({x}_{1}\right),f\left({x}_{2}\right)\right)<\delta$ , then
${d}_{0}\left(f\left({x}_{1}\right),f\left({x}_{2}\right)\right)\le {d}_{\infty }\left({x}_{1}\left(t\right),{x}_{2}\left(t\right)\right)<\delta =ϵ.$

This shows the continuity of $f\left(x\right)$ at ${x}_{1}\left(t\right)$ . However, because the selection of $\delta$ did not depend on the value of ${x}_{1}$ , we have also shown that the function $f$ is uniformly continuous.

## Convergence of functions

Definition 3 The sequence of functions $\left\{{f}_{n}\right\}$ , ${f}_{n}:\left(X,{d}_{x}\right)\to \left(Y,{d}_{y}\right)$ converges pointwise to $f:\left(X,{d}_{x}\right)\to \left(Y,{d}_{y}\right)$ if: for each $x\in X$ , the sequence of values $\left\{{f}_{n}\left(x\right)\right\}$ converges to $f\left(x\right)$ in $\left(Y,{d}_{y}\right)$ .

Definition 4 The sequence of functions $\left\{{f}_{n}\right\}$ , ${f}_{n}:\left(X,{d}_{x}\right)\to \left(Y,{d}_{y}\right)$ converges uniformly to $f:\left(X,{d}_{x}\right)\to \left(Y,{d}_{y}\right)$ if: for each $ϵ>0$ , there exists ${n}_{0}\in {\mathbb{Z}}^{+}$ such that if $n\ge {n}_{0}$ , then ${d}_{y}\left(f\left(x\right),{f}_{n}\left(x\right)\right)<ϵ$ for all $x\in X$ .

The difference between these definitions is that for uniform convergence, there must exist a single value of ${n}_{0}$ that works in the definition of continuity for all possible values of $x\in X$ .

Example 2 Consider the sequence of functions ${x}_{n}\left(t\right):\left(\left[0,1\right],{d}_{0}\right)\to \left(\left[0,1\right],{d}_{0}\right)$ given by ${x}_{n}\left(t\right)=\frac{t}{n}$ . One naturally suspects that ${x}_{n}\left(t\right)$ may be converging to the zero-valued function. We can show this formally as follows:

• Pick some ${t}_{0}$ , and check if $\left\{{x}_{1}\left({t}_{0}\right),{x}_{2}\left({t}_{0}\right),{x}_{3}\left({t}_{0}\right),{x}_{4}\left({t}_{0}\right),...\right\}$ is converging to 0: Denote ${a}_{n}={x}_{n}\left({t}_{0}\right)=\frac{{t}_{0}}{n}$ , and notice that ${d}_{0}\left({a}_{n},0\right)=|{a}_{n}-0|=\frac{{t}_{0}}{n}$ . So, to pick an ${n}_{0}$ such that ${a}_{n}<ϵ$ if $n\ge {n}_{0}$ , one only needs to note that since $\frac{{t}_{0}}{n}\le \frac{{t}_{0}}{{n}_{0}}$ , it then suffices for $\frac{{t}_{0}}{{n}_{0}}<ϵ$ , or in other words ${n}_{0}>\frac{{t}_{0}}{ϵ}$ . So this sequence converges pointwise to 0.
• Additionally, because the range of possible inputs to $x\left(t\right)$ is $t\in \left[0,1\right]$ , we could select ${n}_{0}>\frac{1}{ϵ}$ . Because 1 is the maximum value for ${t}_{0}$ , ${n}_{0}>\frac{1}{ϵ}>\frac{{t}_{0}}{ϵ}$ will work for all values of ${t}_{0}\in \left[0,1\right]$ , and so the sequence $\left\{{x}_{n}\right\}$ also converges uniformly to the zero function.

how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!