\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.

\medskip
(b) Prove that every {\it polynomial function}
$p(x)=a_0+a_1x+\cdots+a_nx^n$ is continuous on $R$.
\medskip
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.

