Let $f(x+y)=f(x)+f(y)\;\text{and}\;\lim_{x\rightarrow 0}f(x)=f(0)$. Prove $f(x)$ is continuous at all $a$.
I want to know if the following substitution I´ve done is a valid step of the proof. I have already seen the other questions related to the problem but my concern is on the specific step.
($f(0)=0$ and $f(-x)=-f(x)$ are implied by de definition of $f(x)$ and will be taken for granted.)
$$0<|u-0|<\delta\Rightarrow |f(u)-f(0)|<\varepsilon$$ $$0<|u|<\delta\Rightarrow|f(u)|<\varepsilon$$
let $u=x-a$ *
$$0<|x-a|<\delta\Rightarrow|f(x-a)|<\varepsilon$$ $$0<|x-a|<\delta\Rightarrow|f(x)-f(a)|<\varepsilon$$ $\square$
Translated to "$\lim$" notation:
$$\lim_{u\rightarrow 0}f(u)=0$$
let $u=x-a$ *
$$\lim_{x\rightarrow a}f(x-a)=0$$ $$\lim_{x\rightarrow a}f(x)-f(a)=0$$ $$\lim_{x\rightarrow a}f(x)=f(a)$$ $\square$
Is the step next to "*" a valid one?