Suppose that a function f is a subadditive prove that if f(0)=0 and if f is continuous at x=0 then f is continuous on all of R ?

how can i solve it ?

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- Apr 24th 2010, 08:37 AMneset44Subadditive Proof
Suppose that a function f is a subadditive prove that if f(0)=0 and if f is continuous at x=0 then f is continuous on all of R ?

how can i solve it ? - Apr 24th 2010, 03:40 PMTikoloshe
A function $\displaystyle f(x)$ is continuous at a point $\displaystyle a$ if $\displaystyle \lim_{x\to a}f(x)=f(a)$. Another characterization of continuity at $\displaystyle a$ (which is useful in your case) is that $\displaystyle \limsup_{x\to a}f(x)-\liminf_{s\to a}f(x)=0$. ($\displaystyle \limsup$ of course is defined as $\displaystyle \limsup_{x\to a}f(x)=\lim_{\epsilon\to0}\ \sup\{f(x):\lvert x-a\rvert<\epsilon,x\neq a\}$

I would use the subadditivity of f along with the above characterization to show continuity at an arbitrary point.