If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$

For $f:\mathbb{R}^n \to \mathbb{R}^m$, if $f(x + y) = f(x) + f(y)$ for then for rational $c$, how would you show that $f(cx) = cf(x)$ holds?

I tried that for $c = \frac{a}{b}$, $a,b \in \mathbb{Z}$ clearly $$ f\left(\frac{a}{b}x\right) = f\left(\frac{x}{b}+\dots+\frac{x}{b}\right) = af\left(\frac{x}{b}\right) $$ but I can't seem to finish it, any help?

Solution 1:

Try computing $bf(x/b){}{}{}$.

Solution 2:

For $c \in \mathbb{Z}^*$:

$$ f(cx) = cf(x) $$

Let's do a variable substitution: $\hat{x} = cx$:

\begin{align*} f(\hat{x}) &= cf\left(\frac{\hat{x}}{c}\right) \\ \Rightarrow f\left(\frac{\hat{x}}{c}\right) &= \frac{1}{c} f(\hat{x}) \end{align*}

Combine this with your initial findings to get the proof for $c \in \mathbb{Q}$.

Solution 3:

Let's show that $f(x/b) = \frac{f(x)}{b}$. Denote $t= f(x/b)$. Now sum $t$ with itself exactly $b$ times:

$$b\cdot t=t+t+\dots+t=f(x/b)+f(x/b)+\dots + f(x/b)$$ $$=f(x/b+\dots+x/b)=f((bx)/b)=f(x)$$

Divide both sides by $b$ and you have $t=\frac{f(x)}{b}$.