Definition 1 (Gnedenko and Kolmogorov (1954), p. 25)

  A frequency distribution F(z) is said to be proper if

$$F(x) - \lim\begin{Sb} h \to 0 \\ h \gt 0 \end{Sb} F (x-h) < 1, \qquad \text{for all } x.$$

The distribution is called improper if

$$F(x) - \lim\begin{Sb} h \to 0 \\ h \gt 0 \end{Sb} F(x-h) = 1, \qquad \text{for some } x, \ \text{say } x_0.$$

We note that in this latter case F(x) = 0 for x < x₀, and F(x) = 1 for $x\geq x_0$.