What is the most painless way to check whether a $f: \mathbb{R}^n \to \mathbb{R}^n$ and $f : \mathbb{R}^n \to \mathbb{R}^{n\times m}$ is continuous Given 
$f(x_1,x_2) = \begin{bmatrix} x_1^2 + x_2 \\ x_1 + x_2^2\end{bmatrix}$
and 
$g(x_1,x_2) = \begin{bmatrix} 2x_1 & 1 \\ 1 &  2 x_2\end{bmatrix}$
In my class I am only taught the $\epsilon-\delta$ method, but oh boy is that method tedious and painful, even for trivial looking functions such as $f(x) = x^{3/2}$ you need divine intervention to fully carry out the proof. There must be a better way, especially given that $f$ and $g$ are obviously continuous elementwise.
What is the "best" way to show that $f,g$ are continuous? 
 A: There are some basic theorems which themselves are proved using the $\epsilon$-$\delta$ method. The way one would approach this problem is to use those theorems (which, of course, have to be proved, but those proofs can be found in any reasonable "advanced calculus" textbook).
First there is a theorem saying that for any function $f=(f_1,f_2)$ with values in $\mathbb{R}^2$, $f$ is continuous if and only if $f_1,f_2$ are continuous. More generally, for any function $f$ with values in $\mathbb{R}^n$ (take $n=4$ for your matrix example), $f$ is continuous if and only if each of its $n$ component functions is continuous. Hence, you need only prove that the various component functions are continuous, namely: 
$$h(x_1,x_2) = x_1^2+x_2 \,\,\text{or}\,\, x_1 + x_2^2 \,\,\text{or}\,\, 2x_1 \,\,\text{or}\,\, 1 \,\,\text{or}\,\,2x_2
$$
Next, there is a theorem saying that constant functions are continuous. This is extremely easy to prove using the $\epsilon$-$\delta$ method. That polishes off the constant function $h(x_1,x_2)=1$, or any other constant function such as $h(x_1,x_2)=2$.
Next, there is a theorem saying that a sum of continuous functions is continuous. Hence, you only need to prove that the various summands are continuous, namely: 
$$h(x_1,x_2) = x_1 \,\,\text{or}\,\, x_1^2 \,\,\text{or}\,\, x_2 \,\,\text{or}\,\, x_2^2 \,\,\text{or}\,\, 2x_1 \,\,\text{or}\,\, 2x_2
$$
Next, there is a theorem saying that a product of continuous function is continuous. Hence, you only need to prove that the various factors are continuous, and having already done so for constant factors, we are down to
$$h(x_1,x_2) = x_1 \,\,\text{or}\,\, x_2
$$
Finally, those last two functions are known as the coordinate projection functions, and there is a theorem saying that the coordinate projection functions are continuous.
Generally speaking, the knowledge of continuity is built up hierarchically, as this example shows. For example, the same basic method that I've outlined above produces the theorem that any polynomial function is continuous, as said in the comment of @PeterFranek.
