What functions $f:\mathbb{R}\to\mathbb{R}$ satisfy $(a+b)f(a+b)=af(a)+bf(b)$?

Question

The relative error transformation $T(f)=\dfrac{f^\prime}{f}$ for differentiable functions $f:\mathbb{R}\to\mathbb{R}$ satisfies the properties

1. $T(fg)=T(f)+T(g)$
2. $T\left(\dfrac{f}{g}\right)=T(f)-T(g)$
3. $T\left(f^n\right)=nT(f)$
4. $(f+g)T(f+g)=f\,T(f)+g\,T(g)$

The first three properties are shared with logarithmic functions on $\mathbb{R}^+$, but not the fourth.

Suppose $f:\mathbb{R}\to\mathbb{R}$ and for $a,\,b\in\mathbb{R}$

\begin{equation} (a+b)f(a+b)=af(a)+bf(b)\tag{1} \end{equation}

Clearly, every constant function defined on $\mathbb{R}$ satisfies this property. If $f(x)=c$ then we have

\begin{eqnarray} (a+b)f(a+b)&=&(a+b)c=ac+bc\\ af(b)+bf(b)&=&ac+bc \end{eqnarray}

Are there other functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy property $(1)$ ?

Answer 1

Let $g(x) = xf(x) \implies g(a+b) = g(a) + g(b)$. The general solution for this functional equation is that $g(x) = cx$. Thus $f(x) = c$ is the only function which meets this condition.