Prove if $f(x)$ and $g(x)$ is continuous, then $f(x) + g(x)$ is also continuous using the $\epsilon - \delta$ definition of limits Since both $f(x)$ and $g(x)$ is continuous, then $$(\forall \epsilon_1 >0)(\exists \delta_1 >0) [\vert x-a\vert< \delta_1 \to \vert f(x)-f(a) \vert <\epsilon_1]$$$$(\forall \epsilon_2 >0)(\exists \delta_2 >0) [\vert x-a\vert< \delta_2 \to \vert g(x)-g(a) \vert <\epsilon_2]$$
I defined $h(x)=f(x)+g(x)$ and for a given $\epsilon>0$, I set both $\epsilon_1,\epsilon_2=\frac12\epsilon$.
Could I ask why would we set $\epsilon_1,\epsilon_2=\frac12\epsilon$? I mean could we can also set it as $\epsilon_1=\frac14\epsilon$ and $\epsilon_2=\frac34\epsilon$?
Additionally, after setting this, how did we derive $\delta=min\{\delta_1, \delta_2\}$, which is required for the proof?

Ok, to see that $h(x)=f(x)+g(x)$ is continuous you have to see that given $\epsilon_{0} >0$ there exist $\delta>0$ such that $|x-a|<\delta$ implies $|h(x)-h(a)|<\epsilon$. Now we know that
$$|h(x)-h(a)|=|f(x)+g(x)-f(a)+g(a)|\leq|f(x)-f(a)|+|g(x)-g(a)|$$
because of the triangle inequality. So we already have our given $\epsilon_{0}>0$ we want to find the $\delta_{0}$. So now we consider $\epsilon_0/2$ (now in a minute you will se why). Since both $f$ and $g$ are continuous, there exit a $\delta_1$ and $\delta_2$ such that 
$$|x-a|<\delta_1 \implies |f(x)-f(a)|<\epsilon_0/2$$
and
$$|x-a|<\delta_2 \implies |g(x)-g(a)|<\epsilon_0/2$$
So if you take $\delta_0=min\{\delta_1,\delta_2\}$ any time $|x-a|<\delta_0$ both of the above will be true. But if both of the above are true, we have that
$$|h(x)-h(a)|\leq|f(x)-f(a)|+|g(x)-g(a)|<\epsilon_0/2+\epsilon_0/2 = \epsilon_0$$ 
And the proposition is proved.
