$BJPHyJ,$N=g=x8r49(B

$i=j$ $B$N$H$-$OL@$i$+$K@.N)$9$k$N$G!"(B$i\ne j$ $B$N>l9g$N>ZL@$,LdBj$H$J$k!#(B

$$\frac{\rd^2 f}{\rd x_i\rd x_j}(a) =\lim_{h\to 0}\frac{1}{h} \left(\frac{\rd f}{\rd x_j}(a+h e_i)-\frac{\rd f}{\rd x_j}(a)\right)$$

$$=\lim_{h\to 0}\frac{1}{h} \left( \lim_{k\to 0}\frac{f(a+h e_i+k e_j)-f(a+h e_i)}{k} - \lim_{k\to 0}\frac{f(a+k e_j)-f(a)}{k} \right)$$

$$=\lim_{h\to 0}\lim_{k\to 0}\frac{1}{hk} \left( f(a+h e_i+k e_j)-f(a+h e_i)-f(a+k e_j)+f(a) \right)$$

$B$G$"$j!"(B

$$\frac{\rd^2 f}{\rd x_j\rd x_i}(a) = \lim_{k\to 0}\lim_{h\to 0} \frac{1}{hk} \left( f(a+h e_i+k e_j)-f(a+h e_i)-f(a+k e_j)+f(a) \right)$$

$B$G$"$k!#6K8B$N=g=x$NLdBj$G$"$k$3$H$,J,$+$k!#

$$\lim_{(h,k)\to(0,0)} \frac{1}{hk} \left( f(a+h e_i+k e_j)-f(a+h e_i)-f(a+k e_j)+f(a) \right)$$

$B$,B8:_$9$k$N$G!"N>

Proof. $i=j$ $B$N>l9g$K@.$jN)$D$3$H$OL@$i$+$J$N$G!"(B$i\ne j$ $B$N>l9g$r>ZL@$9$k!#(B $x_i$ $B$H(B $x_j$ $B0J30$NJQ?t(B $x_k$ ($k\ne i,j$) $B$O!"(B $x_k=a_k$ $B$H8GDj$7$F$$$k$N$G!"K\, $x_j=y$, $a_i=a$, $a_j=b$, $f(a_1,\cdots,a_{i-1},x,a_{i+1},\dots,a_{j-1},y,a_{j+1},\dots,a_{n}) =f(x,y)$ $B$H$7$F!"(B

$$\frac{\rd^2 f}{\rd x\rd y}(a,b) = \frac{\rd^2 f}{\rd y\rd x}(a,b)$$

$B$r<($;$PNI$$!#(B

$$\Delta(h,k):=f(a+h,b+k)-f(a+h,b)-f(a,b+k)+f(a,b)$$

$B$H$*$/(B¹$B!#(B $\phi(x):=f(x,b+k)-f(x,b)$ $B$H$*$/$H!"(B $B==J,>.$5$J@5?t(B $\eps$ $B$r $0<\vert h\vert<\eps$, $0<\vert k\vert<\eps$ $B$rK~$?$9G$0U$N(B $h,k$ $B$KBP$7$F!"(B $\exists\theta,\theta'\in(0,1)$ s.t.

$$\Delta(h,k) =\phi(a+h)-\phi(a)$$

$$=\phi'(a+\theta h)h =\left[f_x(a+\theta h,b+k)-f_x(a+\theta h,b)\right]h$$

$$=f_{xy}(a+\theta h,b+\theta' k)h k.$$

$B$f$($K(B

$$\lim_{(h,k)\to(0,0)}\frac{\Delta(h,k)}{hk} =\lim_{(h,k)\to(0,0)}f_{xy}(a+\theta h,b+\theta' k)=f_{xy}(a,b).$$

$BF1MM$K(B $\psi(y):=f(a+h,y)-f(a,y)$ $B$H$*$/$H!"(B $0<\vert h\vert<\eps$, $0<\vert k\vert<\eps$ $B$rK~$?$9G$0U$N(B $h,k$ $B$KBP$7$F!"(B $\exists\theta'',\theta''\in(0,1)$ s.t.

$$\Delta(h,k)=\psi(b+k)-\psi(b) =f_{yx}(a+\theta'''h,b+\theta''k)hk.$$

$B$3$l$+$i(B

$$\lim_{(h,k)\to(0,0)}\frac{\Delta(h,k)}{hk} =\lim_{(h,k)\to(0,0)}f_{yx}(a+\theta''' h,b+\theta'' k)=f_{yx}(a,b).$$

$B$f$($K(B

$$f_{xy}(a,b)=f_{yx}(a,b). \qed$$

$\qedsymbol$

JN,$9$k!#(B \qed \end{yodan}">

$B1~MQ>e$N4QE@$+$i$O!"(B 2$B2sA4HyJ,2DG=$G$"$k$,!"(B $C^2$ $B5i$G$O$J$$$h$&$J4X?t$,EP>l$9$k$3$H$O5)$J$N$G!"(B $B>e$NM>CL$K=R$Y$?$3$H$r5$$K$9$kI,MW$O$[$H$s$I$J$$!#(B $B$=$l$,8=:_$NHyJ,@QJ,3X$N%F%-%9%H$NB?$/$G!"(B $B9b3,$NA4HyJ,$N35G0$,>JN,$5$l$kM}M3$G$"$m$&!#(B