Polynomial Continuity Proof.

**lllll** · November 9th 2008, 10:41 PM

Prove that the polynomial:

$P(x) = a_nx^n+a_{n-1}x^{n-1}+\dotsc+a_1x+a_0$ is continuous in $\mathbb{R}$

I was thinking of using the Delta Epsilon definition to show this but I couldn't figure out how to actually apply it (in the case of polynomials).

I was also thinking of proving it by induction since I saw a similar case where they showed that $f(x) = x^n$ is continuous. So what I have so far is:

P(1) = a_1x^1 = a_1x which is continuous. ( I don't know if I should show that this actually is or if I can just assume it)

therefore assume that

$P(x) = a_nx^n+a_{n-1}x^{n-1}+\dotsc+a_1x+a_0$ is continuous.

thus

$P(x+1) = a_{n+1}x^{n+1}+a_nx^n+a_{n-1}x^{n-1}+\dotso+a_1x+a_0$

now since I assumed that P(x) is continuous, is the only thing that I have to show is that $a_{n+1}x^{n+1}$ is continuous? If so would the Delta-Epsilon definition suffice?

**Laurent** · November 10th 2008, 12:46 AM

Originally Posted by lllll

Prove that the polynomial:

$P(x) = a_nx^n+a_{n-1}x^{n-1}+\dotsc+a_1x+a_0$ is continuous in $\mathbb{R}$

You probably know that the sum and the product of continuous functions is continuous (if not, prove it, because this is important). As a consequence, the all you have to do is prove that $x\mapsto x$ is continuous, which is obvious from the definition (with $\delta=\varepsilon$ ).

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