A.5.1 $B%9%D%k%`$NDjM}(B

$Be$N6h4V(B $[a,b]$ $B$K$*$1$kB?9`<0(B $f(x)$ $B$N$J(B $BDjM}!#(B

$B!"%&!"%[%HtO}$K$h$C$F!"6h4(B$[a,b]$ $BFb$KB8:_$9$k(B $f(x)=0$ $B$N2r$r?t$rCN$k$3(B $B$H$,$G$-$k$,!"FsJ,C5:w$N5;K!$HAH$_9g$o$;$k$3$H$G!"2r$,B8:_$9$k6h4V$r9%$-(B $B$J$@$1>.$5$/$9$k$3$H$,=PMh!"$=$N6h4VFb$NE,Ev$JE@$r2r$N6a;wCM$H$7$F:NMQ$9(B $B$k$H$$$&6a;w2rK!$,$G$-$k!#$3$l$r(BStrum $B$NJ}K!(B$B$H8F$V!#(B

Proof. $f(x)=0$ $B$N2r$N8D?t$OM-8B8D$G$"$k!#$=$N$&$A(B $[a,b]$ $BFb$K$"$k$b$N$rBg$-(B $B$5$N=g$KJB$Y$F(B

$$x_1<x_2<\cdots<x_{n}$$

$B$H$9$k!#(B$n+1$ $B8D$N6h4V(B

$$I_0=[a,x_1),\quad I_1=(x_1,x_2), \quad I_2=(x_2,x_3), \quad I_{n-1}=(x_{n-1},x_n), \quad I_n=(x_n,b]$$

$B$N9gJ;$K$*$$$F(B $N(x)$ $B$ODj5A$G$-$k!#0J2<$G$O(B

1. $B3F6h4V(B $I_j$ $B$K$*$$$F(B $N(x)$ $B$ODj?t$G$"$k(B:
   $$\exists \{n_j\}_{j=0}^{n}$$   s.t.$$\quad N(x)=n_j$$   $$\mbox{($x\in I_j$, $j=0,1,\cdots,n$)}$$

2. $n_{j+1}-n_j=-1$ ( $j=0,1,\cdots,n-1$)

$B$G$"$k$3$H$r>ZL@$9$k(B ($B$3$NFs$D$+$iMF0W$K(B $N(b)-N(a)=n$ $B$,F3$+$l$k(B)$B!#(B

$\qedsymbol$