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Problem: Find all functions $h:\mathbb{R}\to\mathbb{R}$ such that $$h(x+y) = h(x)+h(y)$$$$h(xy)=h(x)h(y)$$ for all $x,y\in\mathbb{R}$.

My attempt:

From $h(x+y) = h(x) +h(y)$, we can derive that $h(x) = k x$ for all rational $x$ and some rational $k$. Since $h(xy)=h(x)h(y)$, we get $h(x)=x$ for all rational $x$'s. In fact, it is true for all real algebraic numbers, if I argue in a similar manner. However, I cannot extend this to other irrational numbers (Although I have a feeling that I may need to use completeness theorem for $\mathbb{R}$).

Any hints as on how to extend this to all irrational numbers?

Martin Sleziak
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    You can find the full solution here: http://math.stackexchange.com/questions/684078/functional-equations-fxy-fx-fy-and-fxy-fxfy. As for a hint, can you show that $h(x)>0$ for $x>0$? Conclude from that that $h$ is strictly increasing. Then you can use the fact that between every two real numbers there is a rational. – wythagoras Aug 14 '16 at 07:45
  • Thank you so much, @wythagoras – Potemkin Metro Card Aug 14 '16 at 07:55

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