# Prove $|x+y|^{\alpha}\leq |x|^{\alpha}+|y|^{\alpha}$ [duplicate]

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.