Constant Function is of Exponential Order Zero

Theorem

Let $f_C: \R \to \GF: t \mapsto C$ be a constant function, where $\GF \in \set {\R, \C}$.

Then $f_C$ is of exponential order $0$.

Proof

\(\ds \forall t \ge 1: \, \)

\(\ds \size C\)

\(<\)

\(\ds \size C + 1\)

\(\ds \)

\(=\)

\(\ds \paren {\size C + 1} e^{0 t}\)

Exponential of Zero

The result follows from the definition of exponential order, with $M = 1$, $K = \size C + 1$, and $a = 0$.

$\blacksquare$