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For which continuous functions $f:(a,b)\to\Bbb R$ is it true that $\forall x,y\in\Bbb R:f(x+y)\le f(x)+f(y)$ ? Thanks

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For subadditive functions. – user53153 Feb 8 '13 at 22:25
Yeah I was gonna say, what about subadditive functions? This is exactly the definition of subadditivity. Any function with growth less than linear and monotonically increasing for example would work like $\log(x)$ and $\sqrt{x-1}$ on the domain $[1,\infty]$. – Fixed Point Feb 8 '13 at 23:24

Assuming you mean $f : \mathbb{R}\rightarrow\mathbb{R}$, two easy examples are the absolute values, and additive group homomorphisms.

I doubt there's a nice way to describe your class of functions.

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More examples can be obtained by addition: $f(x)=a|x|+bx+c$ with $a\ge 0, c\ge 0$. – Hagen von Eitzen Feb 8 '13 at 22:15
If indeed $f : \mathbb{R}\rightarrow\mathbb{R}$ (and not restricted to some interval only in the positive or only in the negative real numbers)... are there any other examples but those mentioned by @HagenvonEitzen? – mkl Feb 8 '13 at 22:36
@mkl Any (continuous) $f$ with $a|x|+bx+c\le f(x)\le a|x|+bx+2c$ will do: $f(x)+f(y)\ge a|x|+a|y|+bx+by+2c\ge a|x+y|+b(x+1)+2c\ge f(x+y)$ (as above, with $a\ge 0, c\ge0$). – Hagen von Eitzen Feb 9 '13 at 7:35
An interesting family of functions... ;) – mkl Feb 9 '13 at 8:51

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