

      \textbf{4.} Ur�ete $f(x)$, je-li
      $$
            f'(\log x) = \begin{cases}
                  1 & \text{pro } 0<x\le1;\\
                  x & \text{pro } 1<x<+\infty
            \end{cases}
      $$

      \textbf{�e�en�} Spo�teme nejprve $f'(x)$: pokud je $f'(\log x) = 1$, pak i $f'(x) = 1$. D�le,
      pokud je $f'(\log x) = x$, pak $f'(x) = e^x$. Dost�v�me tedy, �e
      $$
            f'(x) = \begin{cases}
                  1 & \text{pro } x\in (-\infty, 0\rangle;\\
                  e^x & \text{pro } x\in (0, +\infty)
            \end{cases}
      $$
      Tedy je
      $$
            f(x) = \begin{cases}
                  x + c_1           & \text{pro } x\in (-\infty, 0\rangle;\\
                  e^x + c_2   & \text{pro } x\in (0, +\infty)
            \end{cases}
      $$
      Aby byla $f(x)$ spojit� polo��me $c_1 = 0$ a $c_2 = -1$.
