I do not understand very well the following text which shows what I ask, if you could help me understand it I would appreciate it.
"We have, then, that if $X$ is a topological space, then $C(X,\mathbb{K})$ is a $k$-algebra of functions. The space $C(X,\mathbb{K})$ contains an isomorphic subbody a $\mathbb{K}$, namely, the space of constant functions. When there is no confusion we will identify the constant functions with the elements of $\mathbb{K}$, so that, for example, $2$ will represent the function that takes the value $2$ over all points of $X$.
Furthermore, if $p(x_1,...,x_n)\in \mathbb{K}[x_1,...,x_n]$, the polynomial p determines a unique evaluation function $p:\mathbb{K}^n→\mathbb{K}$, so that the constant polynomials correspond to the constant functions and the indeterminate $x_i$ to the projection on the $i$-th coordinate. Since every polynomial is a combination of sums and products of constants and indeterminates, we can conclude that $\mathbb{K}[x_1,...,x_n]\subset C(\mathbb{K}^n,\mathbb{K})$"
It is simple: sum and product are continuous functions from $\mathbb{K}^2$ to $\mathbb{K}$, as well as constant functions, projections, and the function $(f,g): \mathbb{R}^n \to \mathbb{R}^m \times \mathbb{R}^k$ for $f: \mathbb{R}^n \to \mathbb{R}^m, g: \mathbb{R}^n \to \mathbb{R}^k$ continuous. Because every polynomial is a combination of compositions of these functions and the composition of continuous functions must again be continuous, polynomials must be continuous.
For example, consider $p(x_1, x_2) = x_1x_2 + x_2^2 \in \mathbb{K}[x_1, x_2]$. Writing $\pi_i$ as the projection projection in the $i$-th coordinate, and $S, P$ as the sum and product, respectively, then we can write the evaluation map $p: \mathbb{K}^2 \to \mathbb{K}$ as $S(P(\pi_1, \pi_2), P(\pi_2, \pi_2))$ - which is a composition of continuous functions.
The argument could be properly arranged into an inductive argument on the degree of the polynomial, however I think it would just make thing more compllicated.