\begin{align*} \forall x_1,x_2 \in \mathbb{Z} \ni x_1 \lt x_2\\ F(x_2) & = \sum_{x \le x_2} f(x) \\ & = \sum_{x \le x_1} f(x) + \sum_{x_1 \lt x \le x_2} f(x)\\ & \ge \sum_{x \le x_1} f(x) = F(x_1) \end{align*}

\begin{align*} \forall x_1,x_2 \in \mathbb{R} \ni x_1 \lt x_2\\ F(x_2) & = \int_{-\infty}^{x_2} f(x) dx \\ & = \int_{-\infty}^{x_1} f(x) dx + \int_{x_1}^{x_2} f(x) dx\\ & \ge \int_{-\infty}^{x_1} f(x) dx\\ & = F(x_1) \end{align*}