I have an example of function which have homogenity but not additive .but now i want function which is Additive but not homogeneity (not linear)
help me thanks.
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The map $f:\mathbb{C} \rightarrow \mathbb{C}$ given by $f(a+ib)= b+ia$ is additive but not homogeneous. For example try comparing $if(i)$ and $f(ii)=f(-1)$.
Over $\mathbb{R}$, things are a lot more complicated. By using the axiom of choice, it can be shown that there do exist functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is additive but not linear. It can be shown that such functions are nowhere continuous and they are unbounded on all non-empty open intervals. For more information about such functions, I'd recommend this wikipedia article on Cauchy's functional equation and this excellent Mathematics Stack Exchange Post.
michaelhowes
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I want example over field R – Cloud JR K Jun 03 '18 at 01:23
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@CloudJR I have edited my answer to include some information about the case when $f$ maps from $\mathbb{R}$ to $\mathbb{R}$. – michaelhowes Jun 03 '18 at 05:53