Mathematics

Problem

Let

f

be a continuous functions satisfying

f'(\ln x) = \begin{cases} 1 & \text{for } 0 < x \le 1 \\ e^x - 1 & \text{for } x > 1 \end{cases}

and

f(0) = 0

, then

f(x)

can be defined as a.

f(x) = \begin{cases} 1, & \text{if } x \le 0 \\ 1 - e^x, & \text{if } x > 0 \end{cases}

f(x) = \begin{cases} 1, & \text{if } x \le 0 \\ e^x - 1, & \text{if } x > 0 \end{cases}

f(x) = \begin{cases} x, & \text{if } x < 0 \\ e^x, & \text{if } x > 0 \end{cases}

f(x) = \begin{cases} x, & \text{if } x \le 0 \\ e^x - 1, & \text{if } x > 0 \end{cases}