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?

Yes, you could also choose $\epsilon_1 = \frac{1}{4}\epsilon$ and $\epsilon_2 = \frac{3}{4}\epsilon$, because you just need
$$
\lvert f(x) + g(x) - (f(a) + g(a))\rvert \leq \lvert f(x) - f(a)\rvert + \lvert g(x) - g(a)\lvert
$$
to be less than $\epsilon$. In fact you just need $\epsilon_1 + \epsilon_2 \leq \epsilon$.
By picking $\delta$ the smallest of the two deltas, you guarantee that both $\lvert f(x) - f(a)\rvert < \epsilon_1$ and $\lvert g(x) - g(a)\lvert < \epsilon_2$.
