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Please explain inequality $|x^{p}-y^{p}| \leq |x-y|^p$
$(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$ for $n$ integer, $n\geq 2$, $W\gt 0$ and $a$ constant, real, $0\lt a\lt 1$

I want to prove that if $0 \leq\alpha\leq1$, then$|x+y|^{\alpha}\leq |x|^{\alpha}+|y|^{\alpha}$for every $x,y$.

In addition, I know that if $f(x)$ is continuous and $f(x+y)=f(x)+f(y)$, then $f(x)=cx$. I want to know what kind of functions $f$ satisfy $f(x+y)\leq f(x)+f(y)$ if $f$ is continuous.


marked as duplicate by Arturo Magidin, Jonas Meyer, azarel, t.b., anon Mar 28 '12 at 4:14

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