# 3.5 Deeper analytic properties of continuous functions

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We collect here some theorems that show some of the consequences of continuity.Some of the theorems apply to functions either of a real variable or of a complex variable,while others apply only to functions of a real variable. We begin with what may be the most famous such result, and this one is about functions of a real variable.

We collect here some theorems that show some of the consequences of continuity.Some of the theorems apply to functions either of a real variable or of a complex variable,while others apply only to functions of a real variable. We begin with what may be the most famous such result, and this one is about functions of a real variable.

## Intermediate value theorem

If $f:\left[a,b\right]\to R$ is a real-valued function that is continuous at each point of the closed interval $\left[a,b\right],$ and if $v$ is a number (value) between the numbers $f\left(a\right)$ and $f\left(b\right),$ then there exists a point $c$ between $a$ and $b$ such that $f\left(c\right)=v.$

If $v=f\left(a\right)$ or $f\left(b\right),$ we are done. Suppose then, without loss of generality, that $f\left(a\right) Let $S$ be the set of all $x\in \left[a,b\right]$ such that $f\left(x\right)\le v,$ and note that $S$ is nonempty and bounded above. ( $a\in S,$ and $b$ is an upper bound for $S.$ ) Let $c=supS.$ Then there exists a sequence $\left\{{x}_{n}\right\}$ of elements of $S$ that converges to $c.$ (See [link] .) So, $f\left(c\right)=limf\left({x}_{n}\right)$ by [link] . Hence, $f\left(c\right)\le v.$ (Why?)

Now, arguing by contradiction, if $f\left(c\right) let $ϵ$ be the positive number $v-f\left(c\right).$ Because $f$ is continuous at $c,$ there must exist a $\delta >0$ such that $|f\left(y\right)-f\left(c\right)|<ϵ$ whenever $|y-c|<\delta$ and $y\in \left[a,b\right].$ Since any smaller $\delta$ satisfies the same condition, we may also assume that $\delta Consider $y=c+\delta /2.$ Then $y\in \left[a,b\right],\phantom{\rule{3.33333pt}{0ex}}|y-c|<\delta ,$ and so $|f\left(y\right)-f\left(c\right)|<ϵ.$ Hence $f\left(y\right) which implies that $y\in S.$ But, since $c=supS,$ $c$ must satisfy $c\ge y=c+\delta /2.$ This is a contradiction, so $f\left(c\right)=v,$ and the theorem is proved.

The Intermediate Value Theorem tells us something qualitative about the range of a continuous function on an interval $\left[a,b\right].$ It tells us that the range is “connected;” i.e., if the range contains two points $c$ and $d,$ then the range contains all the points between $c$ and $d.$ It is difficult to think what the analogous assertion would be for functions of a complex variable, since “between” doesn't mean anything for complex numbers.We will eventually prove something called the Open Mapping Theorem in [link] that could be regarded as the complex analog of the Intermediate Value Theorem.

The next theorem is about functions of either a real or a complex variable.

Let $f:S\to C$ be a continuous function, and let $C$ be a compact (closed and bounded) subset of $S.$ Then the image $f\left(C\right)$ of $C$ is also compact. That is, the continuous image of a compact set is compact.

First, suppose $f\left(C\right)$ is not bounded. Thus, let $\left\{{x}_{n}\right\}$ be a sequence of elements of $C$ such that, for each $n,$ $|f\left({x}_{n}\right)|>n.$ By the Bolzano-Weierstrass Theorem, the sequence $\left\{{x}_{n}\right\}$ has a convergent subsequence $\left\{{x}_{{n}_{k}}\right\}.$ Let $x=lim{x}_{{n}_{k}}.$ Then $x\in C$ because $C$ is a closed subset of $C.$ Co, $f\left(x\right)=limf\left({x}_{{n}_{k}}\right)$ by [link] . But since $|f\left({x}_{{n}_{k}}\right)|>{n}_{k},$ the sequence $\left\{f\left({x}_{{n}_{k}}\right)\right\}$ is not bounded, so cannot be convergent. Hence, we have arrived at a contradiction, and the set $f\left(C\right)$ must be bounded.

Now, we must show that the image $f\left(C\right)$ is closed. Thus, let $y$ be a limit point of the image $f\left(C\right)$ of $C,$ and let $y=lim{y}_{n}$ where each ${y}_{n}\in f\left(C\right).$ For each $n,$ let ${x}_{n}\in C$ satisfy $f\left({x}_{n}\right)={y}_{n}.$ Again, using the Bolzano-Weierstrass Theorem, let $\left\{{x}_{{n}_{k}}\right\}$ be a convergent subsequence of the bounded sequence $\left\{{x}_{n}\right\},$ and write $x=lim{x}_{{n}_{k}}.$ Then $x\in C,$ since $C$ is closed, and from [link]

#### Questions & Answers

Healthcare professionals often work in rotating shifts. Why is this problematic? What can be done to deal with potential problems?
The core problem is the absence Good governance system
Waaya
ONLY BUDGET CONSUMING RATHER THAN WORKING GOOD HEALTH SERVICES FOR COMMUNITY BY LEADERS
Waaya
NO REALITY BUT THERE ARE FALSE BASED REALITY
Waaya
why is stress chronic?
working conditions at work today
working conditions at work
Namwanje
I work in Afghanistan. Work conditions are interesting. 😊
Jonah
what's it like where you work?
Jonah
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queefburgler
thanks young King. Will do. what's goin on out there in you guy's parts of the world?
Jonah
who gets anxiety disorders and how common are they?
I have generalized anxiety and I've had it for almost 12 years
Harley
I've had it for idk how long but I finally went and got diagnosed 3 years ago. it's pretty common actually
Shannon
I have anxiety with secondary depression, basically meaning my anxiety spirals into depression and depressive thoughts
Shannon
depression always start with anxiety......after some year when we are unable to tackle the anxiety....the anxiety converted into depression...b
Sahni
becoz in anxiety a person always unable to taking decision and adjustment with situation et all
Sahni
anxiety big ........ structure is as depression after some times years
Sahni
Do you mind if we begin a conversation?
Not at all, how are you?
Shannon
abt what?
zaynab
I'm totally fine. What about you?
Frank
I'm good, just been studying.
Shannon
Oh,that's pretty good. What have you achieved in your studying?
Frank
what are you guys up to?
queefburgler
I just wanted to have a conversation.
Frank
hitting the books😁👋😂..have a great day everyone 🤗
Michelle
you too Michelle. currently I'm writing a lab report so you could say I'm studying too
queefburgler
Kind of like that. We are having a wonderful day, dude.
Frank
Where are you guys all from?
Frank
I stay in Dubai but originally from Somalia, east Africa
queefburgler
I kind of wanna go to the US after my bachelor's, kinda dont since you guys have that giant orange cheeto thing up in the oval office
queefburgler
But what are you working on?
Frank
currently? I'm doing an organic identification lab report (chemistry)
queefburgler
Unfortunately, i'm really bad at chemistry. Why do you like that? Is there anything that forces you to practice it more and more?
Frank
I'm going to work and then writing a paper for English...we got this...💪😃👍
Michelle
Hi! I'm french
Naomi
I'm from the US, And I wanna leave just because of the orange Cheeto dude
Marco
Oh,Bonjour. I know some words in french.
Frank
Can you help me master that language?
Frank
I'm from Texas currently attending hcc majoring psychology.
Pakayla
and lol Queef.. "the giant orange Cheeto..."
Pakayla
What's the difference between 9/11 and a cow?
Marco
the cow doesnt fly through skyscrapers?
queefburgler
You can't milk a cow for 18 years
Marco
solid.
queefburgler
hi can u eleborate it further?
can I get an explanation on the second requirement which was the use of repeatable stimuli?
what is psyche
psyche is the study of man, man as a being living in the over-world change, responding to thingsand events and other people.
ayman
bhvior n cogntnz....
Hunny
thnkng ptrnz
Hunny
psychology is the study of behavior and mental process
Arooj
something described in the state of a minds being
Carlnisha
That's true
Arooj
YES
ABIOLA
aoa arooj from whr u blong
Hunny
hi
Chandrakant
types of psychology
Clinton
is everyone here belongs to psychology related fields🤔
zaynab
Yes I am
Becca
yea
Odey
yes
JIGNA
behavioural psychology clinical psychology organisational psychology abnormal psychology counselling industrial psychology criminal psychology these all are types of psychology
JIGNA
is any one psychology student ?
JIGNA
i am a psych grad
Gel
Vague to the extent that it sounds like a rhetorical question
keenu
if you want to know the literal meaning of the word, look up Google. Or do provide more info so someone other than you could understand the word in the context that you've come across
keenu
Keenu, what are you talking about?
Becca
Hello I'm anis from Malaysia
Anis
I am strj
Satra
hi
Fred
hello Satra Joy
Fred
Hello
Satra
hey guys
welcome everybody
ayman
u 2
Hunny
am in too
Pauline
Hello
Olukunle
Hello
Dungrani
Tell me some facts about developmental psychlogy
Hasham
what is educational psychology? tell me?
Sonai
What does psyche means? soul
psychology= scientific study of the mind and behavior
Louisiana
Associated to outside stimuli or internal.
Rob
she is a goddess
Elizabeth
it is the study of human nature. it is the study of man, man as a being living in an over-changing world, responding to things and events and other people.
ayman
plz sujeest a book.psychological foundetion of education.
Sonai
Cognition is more about knowing. Learning, for instance, changes or at least has some effect on cognitive style. And Constructivism is more of a method or way to know and learn. This is the way I see these concepts. If you may please, tell me, have I got them correct?
you have to know the rules of life, follow the scientist and their theories. with that you can measure the truth by yourself. take care of you brother
Will
somebody hv any material related to CBT
Hunny
if hv kindly shr me plz
Hunny
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Will
i want to understand by dscsn
Hunny
what is episodic memory?
i don't know
Trish
past memory
Trish
episodic memory is a collection of past personal experiences.
Preet
yes
Nazakat
Try to remember your first day in college(or anywhere). That is your episodic memory. If we discern its characteristics, then obviously its1) Autobiographical, 2)Explicit, meaning it can be expressed verbally, and 3)Retrievable, you can watch that scene on the screen inside your head anytime.
Vijay
that's interesting
Paveethra
yes it is.
mahendra
like watching a season of a show, except it's the show of your life.
Jonah
I just watched the show of my life!!! With deleted scenes(that helps) 🎃
Martin
What does merits mean
worthy or goodneas
Trish
integrity
Genata
merit /ˈmɛrɪt/ ￼Learn to pronounce noun the quality of being particularly good or worthy, especially so as to deserve praise or reward. "composers of outstanding merit" synonyms:excellence, goodness, standard, quality, level, grade, high quality, calibre, worth, good, credit, eminence, worthin
Jonah
so, like a point system in which to grade or determine ones ability to be good or worthy. "Merit System" "Meritorial Award" "...he got it on his merits alone..." etc.
Jonah
look at it like Morale Credit
Jonah
its when credit is given for something you have done
Carlnisha
there is just one definition for psychology ?
Which one are you aware of? I'm aware of that psychology is the study of mind and behaviour?
Onesmo
yea thats the only definition that i found
TheReal
really
ravi
"Scientific" study of behaviour & mental processes..
ravi
Yes of course"scientific study" even the first definition is scientific but I'm wondering if "mental process is not included in the mind cause, mind is all about mental processes!
Onesmo
in general and what they teach in PSYC101 yes, psychology is the scientific study of behavior and mental processes, but as you go deeper, you will get more detailed definitions, for example cognitive psychology is the same definition plus emphasis in mental processes.
Mohammed
ok thanks guys
TheReal
Hlw, everyone. Nice to meet you guys. May I know that How to know about the psychological condition of any person ?
like? you can know based on symptoms
john
actions
Amy
How can I determine, whether the person's intention are not good ?
Sayantan
basically you have to observe any person is he/she anxious or have any stress. for that you have to talk and make a good positive relation and being comfortable to person. you have to assist her personality. don't be judgemental. try to under stand her environment . don't be a emotional.
Nitin
cool, what if my opposite person acts moody....?
Sayantan
if you observe serious mood swings may be he/she suffer from some neurotic problem like bipolar.or may be some action affects negatively
Nitin
observation is very skilled task you can't just learned from a online platform. i just completed my Post graduation diploma in clinical psychology. but still i can't able to well define the mood of patients
Nitin
what's your opinion about the mood swings...? What are the causes (except periods of girls).
Sayantan
I agee on the bipolar, what kind of Moody?
Tina
moody means, suppose a person is talking softly with me but within a fraction of seconds he is starting reacting...
Sayantan
what type of mood swings anger-happy , normal-anger. extra energetic and low aur any other symptom
Nitin
normal- anger that generally causes due to frustration
Sayantan
and due to frustration emerged aggression
Sahni
How many types of characters, does humans have ?
Sayantan
types of characters or should you personality or temperament?
DHRUV
which one suits best ?
Sayantan
if personality, then how many types of personality are their in human
Sayantan
well there 5 basic types defined by terman though there are numerous types as per the different theorists over the time 1. Introversion 2. extraversion 3.agreeable 4. concitiousness 5.neuroticism.
DHRUV
these r 5 traits of personality
DHRUV
Basically I know about the first two. so what are the next 3
Sayantan
im very intrigued
Karena
How are you guys?
Alive and well.
Max
Nice. Where u from?
patwerking
Are u a boy or a girl?
patwerking
pakistan....vehari
Hunny
fin n u
Hunny
Hey
Anna
heloo
Sai
aoa
Hunny
How here can open new topic?
Anna
girl
Hunny
I am from India and u
Sai
want to talk abut negative thoughts abut self about others
Hunny
who
Hunny
ok
Sai
Have anyone worked in the field of mental health?
Gerald
m working at hsptl
Hunny
anyone is listng me
Hunny
hello everyone.. I have completed my master's in clinical psychology and have an internship experience in the field of mental health
Nirali
same here
Hunny
great
Nirali
how r u
Hunny
from where u r?
Hunny
Mumbai, India
Nirali
😯 woow
Khwabeeda
good
Dr
good
Hunny
want to visit india
Hunny
hey there (seniors) ..am an undergraduate student right now
Khwabeeda
oh nice
Nirali
aoa dr azeem
Hunny
from where u r dr azeem?
Hunny
🙂
Hunny
what do I have to do for becoming additional psychologist ? if any of you can suggest some key points
Khwabeeda
I am Ann from Russia. I am 28. :-) I am learning psychology by myself for prepare to enter at the university as a psychologist. I am looking for somebody who want to learn psychology together and discuss topics :-)
Anna
I am 17
Sai
If someone wants to study together and discuss topics, write briefly about yourself and share yours contacts please :)
Anna
I am 17
Sai
i want
Hunny
I am 17
Sai
hello my people
Samuel
m from pakistan...
Hunny
hey
Kanika
i want 2 learn how 2 deal with mently dsturbd prson
Hunny
anna
Samuel
i m 24
Hunny
hy
Hunny
anna....
Hunny
Yeah i want to learn that too
yaswanti
it's very interesting to see there brain function with their illness compare to a some what sane person 🤔
Gerald
how should i deal with somebody who is already too much disturbed with his past... that he doesn't pay attention to his present
yaswanti
Anna how are you
Samuel
also want to learn about brain parts their function n role of neurotransmitters
Hunny
hii
Yendluri
can anyone teach me human psychology
Yendluri
India Punjab Patiala pursuing MAPC IGNU through cc
rajesh
becoz I am going into that field
Yendluri
have u not any idea about human psychology
Hunny
u should go....
Hunny
what do u want to know about human psychology
yaswanti
yaswanti...how can u xplan it
Hunny
explain what...human psychology?
yaswanti
Hunny
completly how people judge and how I have to do it criminal psychology behaviour of humans so on,how to attract people towards our side during interview
Yendluri
a lot of it reiterating that things that have happened in the past is in the past and that we can't go back to change it however we can learn from it so we don't make the same mistakes over and the most important you'll have to find different ways of getting your message a cross.
Gerald
if you know iam glad to know
Yendluri
anyboby have mcqs relatd to psychology with answers for the preparation of compition papr
Hunny
Which exam r u preparing for?
yaswanti
is there anyone to help
Yendluri
4 the admission of ms clinical psychology
Hunny
i nd hlp kindly plz
Hunny
hlw
Sayantan
aoa
Hunny
sorry but i don't know have the MCQs
yaswanti
hmm...its ok dear
Hunny
I want to know how to deal with somebody with a horrible past... somebody full of complexes
yaswanti
Yaswanti Bura, horrible past mean ?
Sayantan
with full trust worthy...
Hunny
no dude , only trust is not the solution
Sayantan
hii. i am kanika from india and i am 18. i have opted for computer science course and want to learn psychology in my part time . can anybody help
Kanika
gee from the state's
Gerald
Horrible past here means...pat full of violence, drug addiction, bad company and loneliness
yaswanti
ok cool, first you have to know whether the person is extrovert or introvert.
Sayantan
according to me, generally people who are extrovert ,they suffer from bad company and loneliness while people who are introvert they goes for violence or drug addictions.
Sayantan
He's definitely an introvert...never shares his problems and acts like everything is fine when from inside he is full of complexes and doesn't trust people
yaswanti
he is addicted to drugs ?
Sayantan
I don't think so
Sayantan
Yes...he says it's a distraction but never mentions distraction from what...though i have a good idea what he's running from
yaswanti
What's your relationship with him ?
Sayantan
We know each other from a year and a half... he's my friend though he claims that I'm his best friend and i know more about him than anyone else
yaswanti
Your and his age ?
Sayantan
16 and...17 respectively
yaswanti
You told , you know lots of things about him than others, right ?
Sayantan
M from Nigeria l am undergraduate student
Zainab
aoa zainab...m sana from pakistan
Hunny
how far
Zainab
Yeah that's right i know more about him than others
yaswanti
I am in skull of community health ,I studies little in spychology
Zainab
Hey I am Tanin! From Bangladesh! I am studying Sociology, BSS Honours First Year..Social Pshychology is one of my major course. I am expecting that you guys will help me 🙂
Shanjida
cool, you know almost more than a half year, so when you first became his friend he was like this only ? or he suddenly changed ?
Sayantan
He has changed...remarkably...he loses his temper more, he more alone, sometimes when he is saying something it's like there's something else he wants to tell me from those lines but i can't grasp the idea, he's more lost now than he was when i met him...
yaswanti
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