Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$, $m≠1$.
I know that $f(x)=x$ and $f(x)=c/(1-m)$ are two solutions.
But to completely solve it I have no idea.
Can we completely solve it using elementary mathematics (without assuming additional conditions) ?

  • $\begingroup$ Are $m,c$ constant? $\endgroup$ – Gina Jul 3 '14 at 5:26
  • $\begingroup$ yes. of course. $\endgroup$ – Bumblebee Jul 3 '14 at 5:29
  • $\begingroup$ Thank you for editing this for me. $\endgroup$ – Bumblebee Jul 3 '14 at 5:32
  • $\begingroup$ Without assuming any extra condition (such as continuity or smoothness) I don't think this is doable with just elementary arithmetic, as it would need some application of Zorn's lemma, or axiom of choice. $\endgroup$ – Gina Jul 3 '14 at 5:39
  • $\begingroup$ @Gina: probably you have in mind using pathological solutions to the Cauchy functional equation $f(x + y) = f(x) + f(y)$. But these don't directly apply because of the part of the condition involving $m$. $\endgroup$ – Qiaochu Yuan Jul 3 '14 at 6:00

Let $g(x)=f(x+\frac c{1-m})-\frac c{1-m}$. Then we have the functional equation $g(mx)=mg(x)$. If $|m|>1$, we can pick any two function $g_0\colon(-|m|,-1]\cup [1,|m|)\to \mathbb R$ and let $$ g(x)=\begin{cases}0&\text{if $x=0$}\\ m^kg_0(m^{-k}x)&\text{if $x\ne0$, $k=\lfloor\log_{|m|}|x|\rfloor$} \end{cases}$$ The case $0<|m|<1$ is equivalent to $g(m')=m'g(x)$ with $|m'|=\frac1{|m|}>1$.

The case $m=-1$ is solved by picking $g_1\colon(0,\infty)\to\mathbb R$ arbitrary and letting $$ g(x)=\begin{cases}0&\text{if $x=0$}\\ g_1(x)&\text{if $x>0$}\\ -g_1(x)&\text{if $x<0$}\end{cases}$$

Remark: We can use arbitrary functions $g_0,g_1$ above. If we wanbt $g$ (and hence $f$) to be continuous for example, we must make sure that $g_0$ resp. $g_1$ is continuous and "fits" at the ends, i.e.

  • $\lim_{x\to \pm m}g_0(x)=mg_0(\pm1)$
  • $\lim_{x\to 0}g_1(x)=0$

Note that the two special solutions you found come from $g(x)=0$ and $g(x)=x$


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