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This question already has an answer here:

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 $$f(x+y)=f(x)+f(y)$$

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marked as duplicate by user228113, user147263, Harish Chandra Rajpoot, Leucippus, user91500 Dec 23 '15 at 4:27

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    $\begingroup$ Try a linear function and see if it works. For instance, take an easy one like $f(x) = 4x$. $\endgroup$ – user137731 Dec 22 '15 at 23:21
  • $\begingroup$ Well, if the function is $\mathbb R\to\mathbb R$, it looks like you just need a $\mathbb Q$-linear map that is not $\mathbb R$-linear. $\endgroup$ – user228113 Dec 22 '15 at 23:25
  • $\begingroup$ thank you sir for your help. $\endgroup$ – Dilzar Bamerny Dec 23 '15 at 0:02
  • $\begingroup$ There are two kinds of "linear", the linear maps, which have the properties you check and linear functions $y = m x + n$ which are affine maps in the context of the first. $\endgroup$ – mvw Dec 23 '15 at 0:04
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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.

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  • $\begingroup$ thank you so much sir. your answer is very helpful. $\endgroup$ – Dilzar Bamerny Dec 23 '15 at 0:14

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