0

What form does a strictly increasing differentiable linear map $f:\Bbb{R}\to\Bbb{R}$ have? Is it necessarily something like $f(x)=\alpha x$ for some real $\alpha$?

vilma
  • 237
  • 2
    You probably want to say "for some $\alpha > 0$" instead of "for some real $\alpha$". Also, any linear function from the reals to the reals has the form $f(x) = \alpha x,$ even if we only assume continuity (even continuity at one or more points; even Lebesgue measurability). Google "Cauchy's functional equation". – Dave L. Renfro Jun 26 '14 at 16:23
  • 1
    What do you mean by linear map? One common definition already implies that $f(x)=f(1)\cdot x$. Or maybe you mean additive functions; i.e., a function fulfilling $(\forall x,y\in\mathbb R)f(x+y)=f(x)+f(y)$? If it is the later, you can find already a few posts about them on this site: http://math.stackexchange.com/questions/423492/overview-of-basic-facts-about-cauchy-functional-equation – Martin Sleziak Jun 26 '14 at 17:02
  • @Martin Sleziak: I read "linear map" as "additive map" by oversight, which you probably noticed. As you said, essentially by definition a linear map has the property the OP is asking about. Maybe the OP meant "additive map". Anyway, I should have discussed both, like you did. By the way, I would upvote your comment, but it seems that most of the time (with rare exceptioins) comment up-voting is disabled for me, and I have no reason why. – Dave L. Renfro Jun 27 '14 at 10:31
  • @DaveL.Renfro My comment was directed to the OP (with the hope that they clarify what they have in mind.) But thanks for your reaction. (Although I expected some kind of reaction from the OP.) – Martin Sleziak Jun 27 '14 at 11:02
  • @Martin Sleziak: For the moment it seems I can upvote comments, so I did it quickly just now, before whatever problem I have reappears. – Dave L. Renfro Jun 27 '14 at 12:02

0 Answers0