real analysis - how can I find a function that satisfies cauchy functional equation?

Can any one help please? I tried to find some linear functions that satisfying $$f(x+y) = f(x) + f(y)$$ but the condition of scalar that says: $$f(ax)=a\cdot f(x)$$ does not hold in cauchy functional equation $$f(x+y)=f(x)+f(y)$$

Answer

This may be what you are looking for. Start with a Hamel basis $H$ (of $\Bbb R$ over the rational field $\Bbb Q$). This is a set $H\subset\Bbb R$ such that every $x\in \Bbb R$ has a unique representation
$x=\sum_{k=1}^n q_kh_k$, where $n\in\Bbb N$, $q_k\in\Bbb Q$ and $h_k\in H$ for $k=1,2,\ldots,n$. Define $f$ arbitrarily on $H$, and then extend $f$ to all of $\Bbb R$ by the recipe
$$
f(x)=\sum_{k=1}^n q_kf(h_k)
$$
in case

$$
x=\sum_{k=1}^n q_kh_k.
$$
The function $f$ so defined satisfies the Cauchy functional equation, but need not be linear.