Constant Function is of Exponential Order Zero
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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 $a = 0$.
$\blacksquare$