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  \section*{Nowhere Differential function}

    �̹��忡���� $[0, 1]$���� $\mathbb{R}$�� ���� nowhere
    differentiable continuous function�� �������� �˾ƺ���.

    (������ �̷��� �Լ����� dense�ϰ� �����Ѵ�. ��, �̷��� �Լ�����
    ������ �Լ������� dense subset�� �Ǹ� ���� ������ �Լ��� �̷���
    �Լ���� �ٻ��ų�� �ִ�.)

    $\mathcal{C}:=\mathcal{C}([0, 1], \mathbb{R})$ with sup metric
    $d(f, g) = \displaystyle{\max_{x\in I} |f(x)-g(x)|}$

    $\Rightarrow$ $\mathcal{C}$�� complete metric space.

    $\Rightarrow$ $\mathcal{C}$�� Baire space.\\

    �־��� $\alp \gg 0$, $0 < h \ll 1$�� ���Ͽ� ������ �����Ѵ�.

    $\displaystyle{\Del f(x, h)= \max(\left| \frac{f(x+h)-f(x)}{h}\right|, \left| \frac{f(x-h)-f(x)}{-h} \right|)}$

    and $$U(\alp, h)=\{f \in \mathcal{C} \mid \Del f(x, h) \geq \alp, \forall x\}$$

    *�鳯 �Լ� $f$�� typical�� ���\\

    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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    $U_n = \bigcup\{U(\alp, h)\mid \alp >n , h < \frac 1 n \}$���
    ����.\\
    
    �׷��� $\displaystyle{f \in \bigcap_{n=1}^{\infty} U_n}$ �̸�
    $f$�� nowhere differentiable function�� �ȴ�:\\

    $\because$ $\forall n$, $f \in U_n$

    $\Rightarrow$ $\forall n$, $\exists \alp_n > n$, $h_n < \frac 1
    n$ such that $f \in U(\alp_n, h_n)$

    $\Rightarrow$ $\forall n$, $\Del f(x, h_n) \geq \alp_n > n$, $\forall x \in I$

    $\Rightarrow$ $\forall x \in I$, $\displaystyle{\lim_{h \rightarrow 0} \Del f(x, h)}$�� ��������
    �ʴ´�.\\


    ���� $f$�� �̺а����ϴٸ�  $\displaystyle{\lim_{h \rightarrow 0} \Del f(x, h)}=|f^{\prime}(x)|$ �� �Ǿ�� �Ѵ�. ���� $f$�� ����������� �̺а������� ������ �� �� �ִ�.\\\\

    ���� $\displaystyle{\bigcap_{n=1}^{\infty} U_n}$�� dense���� ������.
    
    $\mathcal{C}$�� Baire��� �����κ��� $U_n$�� open dense��� ���� ���̸� ������
    ���������� $\displaystyle{\bigcap_{n=1}^{\infty} U_n}$�� dense���� �� ��
    �ִ�.
    
    {\bf Claim 1}. $U_n$ is open:\\

    \textbf{(����)} $f \in U_n \Rightarrow f \in U(\alp, h)$ for some $\alp >
    n$, $h < \frac 1 n$.
    ���� ������ $\epsilon$�� ���� $d(f, g)<\epsilon \Rightarrow g \in U_n$���� ������.

    $\mid \Del f(x,h) - \Del g(x,h)\mid \leq \left| \frac{f(x+h)-g(x+h)-(f(x)-g(x))}{h} \right|
    \leq \frac{2 \epsilon}{h}.$

    ���� $\epsilon = \frac{h(\alp - n)}{4}$���� �����ϸ�
    $\mid \Del f - \Del g \mid \leq \frac{\alp - n }{2}$�ǰ�

    $n < \alp - \frac{\alp - n}2 \leq \Del f - \frac{\alp - n}2
    \leq \Del g \leq \Del f + \frac{\alp -n}2$�� �ȴ�.\\

    {\bf Claim 2}. $U_n$ is dense:\\

    \textbf{(����)} �־��� $g \in \mathcal{C}$, $\epsilon > 0$�� ���Ͽ�
    $d(f, g)<\epsilon$ �� $f \in U_n$�� �������� ������.     
    $diam (g([x_i, x_{i+1}])) < \frac \epsilon 2$�� �����ϴ�
    $[0, 1]$�� partition�� �����Ѵ�.

    �׷��� �� $[x_i, x_{i+1}] \times g[x_i, x_{i+1}]$
    �κ��� $U_n$�� ���� ������ �鳯 �Լ���� ��ü�ؼ� �����ص� $U_n$�� ����. �̶� ���� ����Ʈ�� �鳯�Լ� $f$���� �� ���� �鳯�� ��缱���� ����� �� ���� ���� ���� $\alp$��, �� �� ������ �������� ������ �����ﰢ���� �غ��� $1/2$ ���� ���� $h$�� �����ϸ� $f \in U(\alp, h)$�� �ȴ�.    
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