19 lines
624 B
TeX
19 lines
624 B
TeX
|
\beginsection 17.5
|
||
|
|
|
||
|
|
(a) Prove that if $m\in N$, then the function $f(x)=x^m$ is continuous
|
||
|
|
on $R$.
|
||
|
|
\medskip
|
||
|
|
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.
|
||
|
|
|