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Thread: Polynomial Continuity Proof.

  1. #1
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    Polynomial Continuity Proof.

    Prove that the polynomial:

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

    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 $\displaystyle 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

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

    thus

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

    now since I assumed that P(x) is continuous, is the only thing that I have to show is that $\displaystyle a_{n+1}x^{n+1}$ is continuous? If so would the Delta-Epsilon definition suffice?
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  2. #2
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    Quote Originally Posted by lllll View Post
    Prove that the polynomial:

    $\displaystyle P(x) = a_nx^n+a_{n-1}x^{n-1}+\dotsc+a_1x+a_0$ is continuous in $\displaystyle \mathbb{R}$
    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 $\displaystyle x\mapsto x$ is continuous, which is obvious from the definition (with $\displaystyle \delta=\varepsilon$).
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