Standard Indicator function? Is there any standard function that outputs $1$ if an inequality is true and $0$ otherwise? e.g 
$$F =
\begin{cases}
1 & \text{if }x>D\\
0 & \text{otherwise}
\end{cases}$$
 A: See the Heaviside Step Function defined by
$$\theta(x) = \begin{cases}1, & x \ge 0 \\ 0, & x < 0 \end{cases},$$
with
$$\theta(x-c) = \begin{cases}1, & x \ge c \\ 0, & x < c \end{cases}.$$
A: You've just defined such a function; for your particular condition, this is known as a step function. In the context of probability theory, it's called an indicator function. Some common notations are $\mathbf{1}_{x>D}$ and $[\![ x>D ]\!]$.
A: The problem with using the Heaviside function is that there are several conventions at to what $H(0)$ should equal. Sometimes it is $0$, sometimes it is $\frac{1}{2}$, sometimes it is $1$. To avoid ambiguity, we can write the Kronecker delta $f(x) = H(x) \cdot (1-\delta_{0,x})$; this ensures that $$f(x) =\left\{ \begin{aligned} &0 & x \leq 0 \\ &1 & x > 1 \end{aligned} \right.$$
In the case of the particular example that you gave, this would be $F(x) = H(x-D) \cdot (1-\delta_{D,x})$.
Given all this, I think Yuval Filmus's answer with the indicator function $\Bbb{1}_{x>D}$ is simpler.
