Math Is Fun Forum

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

function

need help with this, thx

I know its easy if you assume f(x) to be n^x, since f(a+b)=f(a)f(b), then find n to be e, but is that enough? I mean it doesnt seem general

I could only get this far

Last edited by Dragonshade (2008-04-30 04:07:41)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: function

says that f is a homomorphism from the additive group of reals to the multiplicative group of positive reals.

says that f is differentiable and hence continuous.

The only continuous homomorphisms from

to

are of the form

where k is a real constant.

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: function

says that f is a homomorphism from the additive group of reals to the multiplicative group of positive reals.

says that f is differentiable and hence continuous.

The only continuous homomorphisms from

to

are of the form

where k is a real constant.

Ah so you can assume its e^kx ? no need to prove that?

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: function

I think it is one of the so called cauchys functions, which does not need to be proven. Jane, can you state the exact theorems about how differentiation and continuity relates to another? didnt know for example that differentiable at a single point, could be extended to all points, i think i need to repeat this

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: function

says that f is a homomorphism from the additive group of reals to the multiplicative group of positive reals.

says that f is differentiable and hence continuous.

The only continuous homomorphisms from

to

are of the form

where k is a real constant.

I believe this is assuming too much for what you are asked to prove.   Not only that, but it's an algebraic proof of an analytic problem.  Certainly this is not what the professor meant.  Furthermore, f'(0) = 1 does not, by itself, say that f is differentiable.  Certainly combined with the other facts it will end up saying that, but it does not follow from what you posted.

Go by definition.  Start out with:

Use the properties that were given in the problem, you should get f(x) almost immediately.

Edit: There is a little "gotcha" in the problem.  Perhaps it's just on my part, but I almost missed this.  At the end of evaluating this limit, you will want to immediately conclude (without further reasoning) that:

Don't.  Be very explicit about it.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: function

Ricky, is my proof under the same principle as your? it looks similar but I dont know if its legal, it goes like this

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: function

No, it's not.  You don't know that f is differentiable everywhere, so f'(x) might not actually exist.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: function

I think I could fix the f(h)/h thing

Last edited by Dragonshade (2008-04-30 09:13:54)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: function

That's it.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."