$BB?JQ?t$NHyJ,@QJ,3X(B1 $BBh(B3$B2s(B

$B7KED(B $BM4;K(B

Date: 2011$BG/(B5$B7n(B9$BF|(B

$B$3$Nhttp://www.math.meiji.ac.jp/~mk/lecture/tahensuu1-2011/

$BA02s$O!"B?JQ?t4X?t$N6K8B$NDj5A$r=R$Y$?!#(B

$BI|=,(B: $BB?JQ?t4X?t$N6K8B(B

$\Omega\subset\R^n$, $\vec f\colon\Omega\to\R^m$, $\vec a\in \overline{\Omega}$, $\vec A\in\R^m$ $B$H$9$k$H$-!"(B

$$\lim_{\vec x\to\vec a}\vec f(\vec x)=\vec A \DefIff \lim_{\vec x\to\vec a}\left\Vert\vec f(\vec x)-\vec A\right\Vert=0$$

$$\Iff \forall\eps>0\quad\exists\delta>0\quad \left(\forall\vec x\i... ...a\right\Vert<\delta\right) \quad\left\Vert\vec f(\vec x)-\vec A\right\Vert<\eps$$

Proof. $1$$BJQ?t4X?t$N>l9g$N>ZL@$r$J$>$k!#(B

($BJL>ZL@$N6Z(B) $B@h$K2CK!(B $(x,y)\mapsto\vec x+\vec y$$B!"%9%+%i!<>hK!(B $(\lambda,\vec x) \mapsto\lambda\vec x$, $BFb@Q(B $(\vec x,\vec y)\mapsto \vec x\cdot\vec y$, $B%N%k%`(B $\vec x\mapsto\left\Vert\vec x\right\Vert$ $B$NO"B3@-$r8@$C$F!"(B $B$=$l$+$i(B $\vec x\mapsto\left(\vec f(\vec x),\vec g(\vec x) \right)$, $\vec x\mapsto\left(\varphi(\vec x),\vec f(\vec x)\right)$ $B$J$I$H$N(B $B9g@.4X?t$H9M$($k!#(B $\qedsymbol$ ARRAY(0xf95d10) $\qedsymbol$

($B:Y$+$$Cm(B: $\dsp\lim_{\vec y\to\vec b}$ $B$,0UL#$r;}$D$?$a$K$O!"(B $\vec b\in\overline V$ $B$G$"$kI,MW$,$"$k$,!"$=$l$O!"(B $\vec f(U)\subset V$, $\vec a\in\overline U$, $\dsp \lim_{\vec x\to\vec a}\vec f(\vec x)=\vec b$ $B$+$i=P$FMh$k!#(B)

Proof. $\forall\eps>0$, $\exists\delta'>0$ s.t.

$$\left\Vert\vec y-\vec b\right\Vert<\delta'\quad\Then\quad \left\Vert\vec g(\vec y)-\vec c\right\Vert<\eps.$$

$\exists\delta>0$ s.t.

$$\left\Vert\vec x-\vec a\right\Vert<\delta\quad\Then\quad \left\Vert\vec f(\vec x)-\vec b\right\Vert<\delta'.$$

$B$3$N$H$-!"(B $\vec y:=\vec f(\vec x)$ $B$H$9$k$3$H$G!"(B

$$\left\Vert\vec x-\vec a\right\Vert<\delta\quad\Then\quad \left\Vert\vec g\left(\vec f(\vec x)\right)-\vec c\right\Vert<\eps. \qed$$

$\qedsymbol$

$BO"B34X?t$,$?$/$5$s$"$k$3$H$r<($=$&!#(B

Proof.

$$\left\vert\varphi_i(\vec x)-\varphi_i(\vec a)\right\vert =\left\vert x_i-a_i\right\vert\le\left\Vert\vec x-\vec a\right\Vert$$

$B$h$jL@$i$+$G$"$k!#0$ --> $\forall\eps>0$ $B$KBP$7$F!"(B $\delta:=\eps$ $B$H$*$/$H!"(B$\delta>0$ $B$G!"(B $\left\Vert\vec x-\vec a\right\Vert<\delta$ $B$J$i$P!"(B

$$\left\vert\varphi_i(\vec x)-\varphi_i(\vec a)\right\vert \le\left\Vert\vec x-\vec a\right\Vert<\delta=\eps. \qed$$

$\qedsymbol$

$n$ $BJQ?t(B $x_1$, $\dots$, $x_n$ $B$N

$$\sum_{i=0}^N\sum_{j=0}^N a_{ij}x^i y^j$$

$B$N7A$r$7$?<0$N$3$H$r8@$&(B ($N\in\N$, $a_{ij}\in\R$)$B!#(B $x_1$, $\dots$, $x_n$ $B$N $\R[x_1, \dots,x_n]$ $B$HI=$9!#(B

$B$^$?(B2$B$D$N $P(x_1,\dots,x_n)$, $Q(x_1,\dots,x_n)$ $B$rMQ$$$F!"(B $R(x_1,\dots,x_n)=\dfrac{Q(x)}{P(x)}$ $B$HI=$5$l$k(B $R(x_1,\dots,x_n)$ $B$N$3$H$r(B $n$ $BJQ?t(B $x_1$, $\dots$, $x_n$ $B$N, $\dots$, $x_n$ $B$N $R(x_1,\dots,x_n)$ $B$HI=$9!#(B

$B $B2f!9$OJ#AG?tCM4X?t$OEvLL9M$($J$$$N$G!"(B $BJ#AG78?tB?9`<0!"J#AG78?tM-M}<0$O07$&I,MW$,$J$$!#(B $BHQ;($5$rHr$1$k$?$a!"!VJ$-!"C1$K!VB?9`<0!W!"(B $B!VM-M}<0!W$H8F$V$3$H$K$9$k!#(B

Proof. (1) $BB?9`<04X?t(B $\vec x\mapsto P(x_1,\dots,x_n)\in\R$ $B$O!"(B $\varphi_i(x)=x_i$ ( $i=1,\dots,n$), $BDj?t4X?t(B $\vec x\mapsto c$ $B$+$i!"(B $BOB$H@Q$r:n$k$3$H$K$h$jF@$i$l$k!#(B $B$f$($K7O(B0.5 $B$K$h$j!"(B $BO"B3$G$"$k!#(B (2) $B$bF1MM$G$"$k!#(B $\qedsymbol$ ARRAY(0xfd7330) $\qedsymbol$

$BB?9`<04X?t!"M-M}4X?t0J30$NO"B34X?t$NNc$O!"(B1$BJQ?t4X?t$K$D$$$F$O!"(B $B$$$/$D$+CN$C$F$$$k!#(B

(a)

$\sqrt{x}$ ($x\ge 0$)

(b)

$\alpha\in\R\setminus\Z$ $B$KBP$9$k(B $x^\alpha$ ($\alpha>0$ $B$N$H$-$O(B $x\ge 0$, $\alpha<0$ $B$N$H$-$O(B $x>0$)

(c)

$e^x$

(d)

$\log x$ ($x>0$)

(e)

$\cos x$, $\sin x$

(f)

$\tan x$ ( $x\in\R\setminus\{(n+1/2)\pi; n\in\Z\}$)

$B$3$l$i$N4X?t$NO"B3@-$O4{CN$H$9$k!#(B