Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function 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?

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

$$f_\epsilon(x) = f_1(x/\epsilon) = \begin{cases} 0 & \text{ if } x < 0 \\ 1 & \text{ if } x/\epsilon \ge 1 \end{cases}$$ $$\left|\frac{d^k}{dx^k}f_\epsilon\right| \le \frac{C}{\epsilon^k}$$ 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?

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

$$f_\epsilon = \begin{cases} 0 & \text{ if } x < 0 \\ 1 & \text{ if } x \ge \epsilon \end{cases}$$ $$|\frac{d^k}{dx^k}f_\epsilon| \le \frac{C}{\epsilon^k}$$ for $k \in \{1,2,\dots,m\}$ and some constant $C>0$, and $$\int_{0}^{1} |\frac{d^k}{dx^k}f_\epsilon| dx \le \int_0^1 |f_\epsilon| dx$$

hold?

Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function 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?