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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.

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Use that $x\mapsto const$, $x\mapsto x$ are continuous and that sum and product of two continuous functions are continuous. –  Hagen von Eitzen Jan 22 '13 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. –  Tunococ Jan 22 '13 at 12:52
    
@HagenvonEitzen, if you write that as answer, i will accept and upvote it. and Thank You. –  007resu Jan 22 '13 at 13:20
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The coefficients $a_n$ of a polynomial are constant in the sense that, even though they could be any number, they cannot depend on $x$. –  Marc van Leeuwen Jan 22 '13 at 13:20

1 Answer 1

up vote 4 down vote accepted

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.

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