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\begin{document}
\begin{center}
{\large ���wI���K�@��W��@2007�N10��16���z�z} 

\smallskip

�S�� ���n����CTA �O�p �~
\end{center}

\smallskip
{\small

���K����
{\tt http://www.ms.u-tokyo.ac.jp/$\sim$hirachi/courses/sugaku-I-2007/}
����_�E�����[�h�ł��܂��D�u�`�������ڂ��Ă��܂��D
}


\bigskip

{\bf ���.} $\R$��̔����\�Ȋ֐�$f$�ɂ�������$f'$���L�E�ł����
$f$�͈�l�A���ł��邱�Ƃ������B


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{\small
�ȉ��̖����ł���͈͂ŉ���, 10��23��13���܂łɃA�h�~�j�X�g���[�V�������̃��|�[�g��o�{�b�N�X�ɒ�o���邱��. 
}

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\prob 
$A,B$����ɗL�E��$\R$�̕����W���Ƃ���.�ȉ��̂��Ƃ�����.

(1)$\sup\{ax\ | x\in A\} = a\sup A\qquad a>0$ 

(2)$\sup\{x+y\ | x\in A, y\in B\} = \sup A +\sup B$

(3)$-\inf\{-x\ | x\in A\}=\sup A$


\prob
  $f$�𔼊J���$[a,b)$��ň�l�A���Ȋ֐��Ƃ���D���������D


(1) $b$�Ɏ�������$[a,b)$���̐���$\{x_n\}_{n=1}^\infty$�ɑ΂���
$\{f(x_n)\}_{n=1}^\infty$��Cauchy��ł���D
�i�����$\lim_{n\to\infty} f(x_n)$�����݂���D�j

(2) $b$�Ɏ�������$[a,b)$���̔C�ӂ̐���$\{y_n\}_{n=1}^\infty$�ɑ΂�
$$\lim_{n\to\infty} f(x_n)��\lim_{n\to\infty} f(y_n).
$$

(3) $b$�ł̍��Ɍ�$\lim_{x\to b-0}f(x)$�����݂���.





\prob
���̂��ׂĂ̏����𖞂���$\R$��̎����l�֐�$f(x)$�͑��݂��邩�D
�������݂���Ȃ�C�������,���̗Ⴊ�ȉ��̏������m���ɖ������Ă��邱�Ƃ��ؖ�����D���݂��Ȃ��Ȃ�΁C
���݂��Ȃ����Ƃ��ؖ�����D

(1) $f(x)$��$\R$���l�A���ł���D

(2) $f(x)$��$\R$������\�ł���D

(3) �W��$\{f'(x)\mid x\in \R\}$�͗L�E�ł͂Ȃ��D

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