\beginsection 17.5

(a) Prove that if $m\in N$, then the function $f(x)=x^m$ is continuous
on $R$.
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The function $f(x)=x$ is continuous. Theorem 17.4 (ii) tells us that the product
of continous functions is continuous.
Since $x^m$ is the product of $m$ continuous functions, $x^m$ is continuous.

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(b) Prove that every {\it polynomial function}
$p(x)=a_0+a_1x+\cdots+a_nx^n$ is continuous on $R$.
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By Theorem 17.4 (ii) and Exercise 17.5 (a) above, each term in the polynomial
is continuous.
By Theorem 17.4 (i), the sum of the terms in the polynomial is continuous.
Therefore, $p(x)$ is continuous.