Continuity of the extended exponential function

Question

I want to show that the extended exponential function $\text{Exp}:\mathbb R \cup \{\pm \infty\} \to \mathbb R \cup \{\pm \infty\}$, $$\text{Exp($x$)} = \begin{cases}\exp(x) & x\in \mathbb R\\0 & x = -\infty \\ +\infty & x= +\infty \end{cases} $$ is continuous. I know that $\exp(x)$ is continuous on $\mathbb R$. I could show it pretty easily using the sequence criterion for continuity: Let $(a_n)_{n\in \mathbb N}$ a sequence in $\mathbb R$ with $\lim_{n\to \infty} a_n = +\infty$. It easily follows that, since $\exp(x)$ is continuous, that $$\lim_{n\to \infty} \exp(a_n) = \exp(\lim_{n\to\infty} a_n) = \exp(+\infty) = +\infty$$ (analogous for $x = - \infty$). However, I tried it directly with the $\epsilon-\delta$-criterion too but couldn't get far since I don't know how for example the expression $|x-(-\infty)|<\delta$ is to be interpreted. Could someone give me a proof or an ansatz for it?