# Prove that any polynomial function is continuous

I am expected to prove by induction that any polynomial function is continuous. In which "direction" would you advise to make induction?

e.g. Taking $x^n$ and making induction on $n$ is not sufficient. By polynomial I understand, $\sum^{m}_{n=1}a_n x^n$. How do I prove it's continuity using epsilon delta notation? And further do I make an induction on $n$ only? Is not $a_n$ a variable as well?

It seems complicated. If you have any suggestion, I would be obliged.

• Use that $x\mapsto const$, $x\mapsto x$ are continuous and that sum and product of two continuous functions are continuous. Jan 22, 2013 at 12:49
• If you allow $a_n$ to vary as well, then the epsilon-delta definition might be a bit too fundamental to use... I think when people speak of polynomials, they usually don't allow $a_n$ to vary. Anyway, your function will still be continuous even though you think of $a_n$ as variables. Jan 22, 2013 at 12:52
• @HagenvonEitzen, if you write that as answer, i will accept and upvote it. and Thank You. Jan 22, 2013 at 13:20
• The coefficients $a_n$ of a polynomial are constant in the sense that, even though they could be any number, they cannot depend on $x$. Jan 22, 2013 at 13:20

You should already know the following functions are continuous:

• $x\mapsto const$
• $x\mapsto x$
• $x\mapsto f(x)+g(x)$ where $f,g$ are continuous
• $x\mapsto f(x)\cdot g(x)$ where $f,g$ are continuous

Every polynomial function can be obtained from these in finitely many steps.

Induction base step: Constant functions are continuous.

Proof: Recall the (Weierstrass) definition of continuous function. For every $\epsilon$ there is a $\delta$ such that for $x_0 − \delta < x < x_0 + \delta$ implies $f(x_0)-\epsilon < f(x) < f(x_0)+\epsilon$. This is easy. Take $\delta$ to be one (or a million).

Induction hypothesis implies inductive step: Next, you need to assume that polynomials of degree $n-1$ to be continuous, then show that a polynomial of degree $n$ is also continuous.

Here is what you need to show 1) $f(x)=x$ is continuous. This is easy. For any $\epsilon$ take $\delta$ to be equal to $\epsilon$. 2) the sum of continuous functions is continuous, and 3) the product of of two continuous functions is continuous. These last two results are in any calculus book.

With this, simply note that any polynomial of degree $n$ equal to $p_n(x)$ can be written as $p_{n-1}(x)\cdot x+constant$. (The constant is $p_n(0)$.)