nLab
polynomials are continuous

Statetement

Given a polynomial $P \in \mathbb{R}[X]$ over the real numbers, we may regard it as a function $P \colon \mathbb{R} \longrightarrow \mathbb{R}$. If here $\mathbb{R}$ is regarded as a Euclidean space (equipped with its metric topology), then this is a continuous function.

Related concepts

• rational functions are continuous

References

The proof using epsilontic analysis is spelled out for instance in

• Kyle Miller, Polynomials are continuous functions, 2014 (pdf)