Corollary 6

  If for a sequence of constants a_n > 0 and b_n, $n = 1,2,3,\ldots$,

$$G_{m,n} (a_n x + b_n) \Rightarrow H_m (x),$$

where m is finite and H_m (x) is a proper frequency distribution, then H_m (x) is everywhere continuous.

Proof:

$$\text{If} \quad G_{m,n} (a_n x + b_n) \Rightarrow H_m(x), \quad \text{then} \quad F^n (a_n x + b_n) \Rightarrow \phi (x),$$

by Theorem 6. But by Corollary 4, $\phi(x)$ is everywhere continuous. In the proof of Theorem 6 it was shown that the set of continuity points of H_m(x) and $\phi(x)$ are identical. Hence, H_m(x) is everywhere continuous.