polynomials are continuous

**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

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…

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.

The proof using epsilontic analysis is spelled out for instance in

- Kyle Miller,
*Polynomials are continuous functions*, 2014 (pdf)

Last revised on May 27, 2020 at 08:29:00. See the history of this page for a list of all contributions to it.