Background
I have seen proofs showing that the sums of two continuous functions $f_1, g_1 :$$\mathbb{R} \rightarrow \mathbb{R} $ are continuous, and I have also seen this result for functions $f_2, g_2 :$$\mathbb{R}^n \rightarrow \mathbb{R}$.
However, I have not seen a generalisation of the claim for when we take the functions $f_3 : \mathbb{R}^k \rightarrow \mathbb{R}^n$ and $g_3 : \mathbb{R}^q \rightarrow \mathbb{R}^n $. If both of these functions are continuous, then this should imply that the sum is also continuous, however, I am struggling to prove the claim.
For clarity, we define the sum to be the function $h(x,y) := f_3(x) + g_3(y)$, where we want to prove that this is continuous given that $f_3, g_3$ are continuous.
Attempt
I have shown that the projections of $ \mathbb{R}^k $ and $ \mathbb{R}^q $
$p_1(x,y) := x \space \space$ and $\space \space p_2(x,y) := y$
are both continuous. But am unsure of how to proceed from here.
My initial thoughts are that an epsilon - delta argument should be able to work for two specified values of delta which are valid for showing that the individual functions are continuous (by our assumption).
I’m assuming we can then use these to construct a new values of delta that will hold for the sum.
However, I haven’t made much progress here as of yet.
If anyone could help me construct a proof, or point me towards a reference, I would be grateful.