Re: [math-fun] Continuous derivations on the reals

Let [1] d(uv) = d(u) v + u d(v) Then [2] d(u^2) = 2u d(u) If d has a Maclaurin series, [2] implies that its coefficients are all 0, so d = 0 within its radius of convergence. My real analysis is rusty, I'm not sure what this implies about the original question. ----- Original Message ----- From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Sunday, June 17, 2007 12:58 PM Subject: [math-fun] Continuous derivations on the reals

But what are the continuous derivations d: R -> R ?

I.e., for all real u,v, we must have

d(uv) = d(u)*v + u*d(v)

The 0 function works. What is the general solution?