Function of Exponential Order of Scalar Multiple

Theorem
Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$.

Let $\lambda$ be a real constant.

Let $\map f t$ be of exponential order $a$.

Then the function defined by $t \mapsto \map f {\lambda t}$ is of exponential order $a\lambda$.

Proof
The result follows by the definition of exponential order.