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I'm confused as to go about this problem. I feel as if we have to show that

$P [0,1] \in C^{0}[0,1]$

by letting

$f = a_{n}x^{n} + a_{n-1}x^{n-1} + .... + a_{1}x^{1} + a_{0}$

  1. We must show that if $f,g \in P([0,1])$, then $f+g \in P([0,1])$
  2. Show that $a_{n}x^{n} \in C^{0}[0,1]$

Can anyone help me out where to go from here?

user77107
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  • Note that it suffices to prove that $x^n$ is continuous on $[0,1]$, by linearity (which is easy to prove directly). To prove continuity for $x^n$, you might want to induct on $n$, or prove that the product of two continuous functions is continuous (which might become overly-general). – awwalker May 15 '13 at 22:30
  • You can not use that sums and products of continuous functions are continuous? – Sigur May 15 '13 at 22:30
  • I don't understand the point of the first question. Shouldn't you replace $P([0, 1])$ by $C^0[0, 1]$ in this question? You can then combine both questions to show the desired result. (What your version of the first question asks you to prove is true but I fail to see where you will need to use it.) – a3nm May 15 '13 at 22:31
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    Without quoting any theorems? – Thomas Andrews May 15 '13 at 22:46
  • by theorem they are continuous! no theorems quoted! However try $\epsilon-\delta$-definition to prove that a polynimial is continuous at $0$ (not so hard). then it's easy to generalize the proof to any point. –  May 15 '13 at 22:54

1 Answers1

4

Steps:

  • Prove that constant functions are continuous
  • Prove that the identity function $f(x)=x$ is continuous
  • Prove that if $f(x)$ and $g(x)$ are continuous, then so are $f+g$ and $f\cdot g$.

This suffices to prove that all polynomials are continuous.

Thomas Andrews
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    For $f\cdot g$, use the following trick: $$f(x)g(x)-f(x_0)g(x_0)\ =\ f(x)g(x)-f(x)g(x_0)+f(x)g(x_0)-f(x_0)g(x_0),.$$ – Berci May 15 '13 at 22:51