Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2017-07-25 05:00:37

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,843
Website

### Functional Equation

Let f be a continuous function on the reals. Furthermore, assume that f is differential everywhere.

For all real x and y (such that y != 0), the following holds: f(x-y) = f(x)/f(y)

Assume that f'(0) = p

Can we use this to determine f?

Last edited by Agnishom (2017-07-25 05:03:31)

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline

## #2 2017-07-25 05:35:23

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,154
Website

### Re: Functional Equation

I would say yes: taking
gives you
, and taking
gives you
. Use the definition of the derivative together with this relation to arrive at
. That's a first order ODE whose solution you can uniquely determine due to the initial condition.

Online

## #3 2017-07-25 13:52:36

Agnishom
Real Member
From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,843
Website

### Re: Functional Equation

Oh yes, that totally works. Thank you

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Offline