Math Is Fun Forum

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

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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)

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'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.

zetafunc

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Registered: 2014-05-21

Posts: 2,436

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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.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: Functional Equation

Oh yes, that totally works. Thank you

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'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.