Proof

Due to example and fact, the powers

- $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

are continuous for every ${}n\in \mathbb {N}$. Hence, also the functions

- $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax^{n},$

are continuous for every ${}a\in \mathbb {R}$ and therefore, again due to fact, the functions

- $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},$

are continuous.