Is there a $C^m$ approximation $f_\epsilon$ of the [Heaviside function][1] such that 

$$f_\epsilon = \begin{cases} 0 & \text{ if } x < 0 \\
1 & \text{ if } x \ge \epsilon
\end{cases}$$
$$\left|\frac{d^k}{dx^k}f_\epsilon\right| \le \frac{C}{\epsilon^k}$$
for $k \in \{1,2,\dots,m\}$ and some constant $C>0$, and 
$$\int\limits_{0}^{1} \left|\frac{d^k}{dx^k}f_\epsilon\right| dx \le \int\limits_0^1 |f_\epsilon| dx$$

hold?

  [1]: https://en.wikipedia.org/wiki/Heaviside_step_function