Derivative and Continuity of the Scaled Exponential Function

Let $\mathbb{R}$ be the real numbers, let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, let $\exp$ be the exponential function, let $c\in\mathbb{R}$, and define $E_c:\mathbb{R}\to\mathbb{R}$ by

$$E_c(t)=\exp(ct)\qquad(t\in\mathbb{R}).$$

The set $\mathbb{R}$ is an interval, and every $t\in\mathbb{R}$ is an interior point of it, since $t-1,t+1\in\mathbb{R}$ satisfy $t-1<t<t+1$ by claims 6 and 1 of Elementary Order Arithmetic in an Ordered Field.

Then the following hold.

1. (Derivative) $E_c$ is differentiable at every $t\in\mathbb{R}$, with

$$E_c'(t)=c\,\exp(ct).$$

2. (Continuity) $E_c$ is continuous on $\mathbb{R}$, regarded as a map from $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$.