Basic Properties of the Exponential Function

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\exp:\mathbb{R}\to\mathbb{R}$ be the exponential function and let $e=\exp(1)$ be as in The Real Exponential Function (iii).

(i) $\exp(0)=1$.

(ii) $\exp(u+v)=\exp(u)\exp(v)$ for all $u,v\in\mathbb{R}$.

(iii) $\exp(u)>0$ for every $u\in\mathbb{R}$.

(iv) $\exp(-u)=1/\exp(u)$ for every $u\in\mathbb{R}$, the quotient being defined by (iii).

(v) $\exp(u)\ge1+u$ for every $u\in\mathbb{R}$.

(vi) $|\exp(h)-1-h|\le e\,h^{2}$ for every $h\in\mathbb{R}$ with $|h|\le1$.

(vii) For every interval $I$ containing two distinct points, the restriction $\exp|_{I}:I\to\mathbb{R}$ is differentiable, and its derivative at every $x\in I$ is $(\exp|_{I})'(x)=\exp(x)$.

(viii) $\exp$ is continuous on $\mathbb{R}$, as in The Real Line: Standing Notation and Background for Calculus on Intervals (vi).

(ix) $\exp(u)<\exp(v)$ for all $u,v\in\mathbb{R}$ with $u<v$.

(x) For every real $\varepsilon>0$ there is $M\in\mathbb{R}$ such that $\exp(u)<\varepsilon$ for every $u\in\mathbb{R}$ with $u<-M$.

(xi) The map $\mathbb{R}\to(0,\infty)$, $u\mapsto\exp(u)$, into the ray $(0,\infty)$ (a map by (iii) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), is bijective.