Question

All functions that satisfy continuity and smoothness have a derivative. Do all functions that satisfy those conditions have an anti-derivative? Why or why not?

Answer 1

yes they do. the anti- derivative is the integral

Answer 2

can you give a better explanation cus i need to explain

Answer 3

Your first statement is something of a tautology. If a function is 'smooth' at a point or on an interval or whatever, then we mean it has a derivative on that set.

Answer 4

true

Answer 5

continuity implies that you have both a derivative and an integral at that point. consider an integral to be the area under the smooth curve, if the curve is continuous then there exists area underneath it.

Answer 6

Yeah Agree

Answer 7

of course it might not be in a nice closed form, but the fundamental theorem of calculus say "the derivative of the integral is the integrand" so if you have an integrable function
\[f(x)\]then
\[\int_a^xf(t)dt\] has derivative
\[f(x)\]

Answer 8

But I will say this. The conditions for smoothness; i.e., the ability to differentiate are stricter than those to integrate, i.e., the anti-derivative exists.

I.e., there are functions which are not differentiable that are (Riemann) integrable.

(@moses: continuity most definitely does not imply differentiability. For example f(x) = |x|. This function is continuous everywhere but not differentiable at x = 0)

Answer 9

well since the function isn't smooth

Answer 10

it could be smooth (like gary coleman) or rough, but if it is continuous it is integrable. so it has an "anti - derivative"

Answer 11

that point is not differentiable at x=0 true, but that is a singularity or some sharp point and is not smooth...

Answer 12

smooth = differentiable. That's what smooth means.

Anyway, what sat73 said: continuous functions are always integrable. But as we just saw, continuous functions aren't always differentiable.

Answer 13

Ok thanks guys for your help ;D

Answer 14

so why cant you differentiate at x=0 is the tan line fails the horizontal line test? or what? How does it work JamesJ?

Answer 15

Singularity sounds cool
What is Singularity in relation to Black Hole?

Answer 16

derivative is a limit. not differentiable means limit does not exist

Answer 17

For f(x) = |x|, the derivative at zero is defined as the limit as a --> 0 of the difference quotient: (f(a) - f(0))/(a-0) = |a|/a.

Now it's clear that limit doesn't exist; from the left it is -1; but from the right it is +1.

Therefore f is not differentiable at x = 0.

Answer 18

Jamesj then why is the limit taken as a->constant = 0 a bad thing? why do we consider it impossible?

Answer 19

I don't quite understand what you're asking. But a necessary condition for any limit on the real line to exist is that the left-hand limit equal the right-hand limit. In this case it doesn't and therefore the limit doesn't exist.

But setting all of that analysis aside, just look at the graph of f(x) = |x|.

It's really no surprise that the function doesn't have a derivative at x = 0, as what could it possibly be. Get arbitrarily close to x = 0 from the negative side and the tangent line has slope -1. But from the positive side it is +1. How can we draw a unique tangent line at 0? We can't.

Answer 20

what implies uniqueness when drawing tangent lines?

Answer 21

Technically, the existence of the limit.

Answer 22

james are u a mathmetitian?

Answer 23

yes

Answer 24

That is what it sounds like!!! You have a great understanding of concepts and can explain them clearly

Answer 25

How'd you do it, James How did you get so good at math?

Answer 26

how do i prove an existence of a limit? maybe i need some help with the theory of limits. eps delta proof?

Answer 27

Well u take a limches a from the right hand side and left hand side and if they equal each other than there is a limit

Answer 28

* as x approaches A

Answer 29

hard work mostly: do a lot of examples, and what Paul Halmos said:

"Don't just read [the proof]; fight it!

"Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"

Answer 30

ok guys. off to watch Top Chef now. Bye.

Answer 31

Cool! Thanks
I hope I can get as motivated as you are

Answer 32

thanks bye

Answer 33

Bye

Answer 34

My dad want sme to become one!!!!!

Answer 35

I guess my explanation wasn't good enough

Answer 36

I asked this question b4 I approached u

Answer 37

I posted the exact words u told me

Answer 38

Okay, I guess nobody's explanation was good enough. You clearly have tough standards. James will be shocked that you chose my explanation over his.

Answer 39

I dont think he wld care less

Answer 40

I think he would care, because he's a caring person.

Answer 41

Ya but he wldnt care if i used s/o elses explanation over his?!?1?
Y wld he care

Answer 42

If s/o is really good at s/t they know it and dont need others approval

Answer 43

rld, you seem so gullible at times, lol

it's cute

Answer 44

oh u were joking???

Answer 45

And funniest part about the whole thing is you asking that question.

Answer 46

I definately cant figure u out!!!

Answer 47

Oh trust me. I would never task anyone with trying to do that. Sometimes, I can't even figure myself out.

Answer 48

That is a problem!!!

Answer 49

I don't disagree

Answer 50

Anyway, I'm not going to hold up anymore of your time. I'm sure you should be getting back to work now :P

Answer 51

ya i shld be. so far behind as usual

Answer 52

Besides, I could be watching Smallville right now.

Answer 53

go do that

Answer 54

that wld be educational

Answer 55

Please...if you took one course at my university, you would die by day 3

Answer 56

i dont think so. i wld stick it out

Answer 57

i just need e/t to be perfect

Answer 58

You're too used to finishing courses in two months. If you came here, the course would be done and you would be asking everybody why there's no school tomorrow

Answer 59

LOL

Answer 60

I wld be like i dont even know what i learnt

Answer 61

Goodnight rld :P

Answer 62

Not going to bed yet!!!

Answer 63

me neither but I'm getting off of here. I can't believe I'm addicted to a nerd site.

Answer 64

ya u r

Answer 65

What do ur freinds think that u sit on here all day?????

Answer 66

Nobody knows what I do here. I live in a private dorm

Answer 67

oh cool. hey u r leaving that place or r u?

Answer 68

I would never tell them. That would be too embarrassing

Answer 69

I'm never leaving this town. This is my home

Answer 70

Sorry to disappoint you

Answer 71

y wld i be dissappointed???????

Answer 72

lol, idk

Answer 73

It was just a figure of speech

Answer 74

Okay gn

Answer 75

yup gn

Answer 76

go watch smallville