# Proving that all polynomials are continuous

Prove that all polynomials from $\mathbb{R}$ to $\mathbb{R}$ are continuous. Now this is from a topological point of view.

I thought that maybe induction would work here?

Initial Case: $f(x) = a_{0}$ where $a_{0} \in \mathbb{N}$. This is continuous.

Inductive Case: $f(x) = a_{0} + a_{1}x + a_{2}x^2 + ... + a_{n}x^n + ...$

Since the sum of continuous functions is continuous, this implies every term is continuous. Makes sense. Feel like theirs a lot of holes in my reasoning though. For one I am not sure how to show the constant $a_{0}$ is continuous from a topological view.

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The constant function is particularly easy: the inverse image of any open set $S$ is either the empty set or $\mathbb{R}$, depending on whether $a_0\in S$. Both of these are open. – user7530 Feb 26 '13 at 3:48

$$p_0(x) = a_n$$ $$p_1(x) = x p_0(x) + a_{n-1}$$ $$p_2(x) = x p_1(x) + a_{n-2}$$ $$\dots$$ $$p_n(x) = f(x)$$