Termial on Real Numbers is Extension of Integers

Theorem
The termial function as defined on the real numbers is an extension of its definition on the integers $\Z$.

Proof
From the definition of the termial function on the integers:
 * $\ds n? = \sum_{k \mathop = 1}^n k = 1 + 2 + \cdots + n$

From Closed Form for Triangular Numbers, we have that:
 * $\ds \forall n \in \Z_{> 0}: \sum_{k \mathop = 1}^n k = \dfrac {n \paren {n + 1} } 2$

This agrees with the definition of the termial function on the real numbers.

Hence the result, by definition of extension.