Continuity between two metric-spaces

Question

How to prove (without sequences) that $id: (\mathbb{R},|.|)\to (\mathbb{R},d)$ is continuous?

where $d(x,y)=|\exp(x)-\exp(y)|$

can we say: id is continuous iff $$\forall x_0 \in \Bbb{R}, \forall \varepsilon>0,\exists \delta>0, \forall x\in \mathbb{R}; |x-x_0|<\delta\Rightarrow |\exp(x)-\exp(x_0)|<\varepsilon$$ ? but how to obtain this?

Answer 1

The condition in question is nothing but continuity of the exponential function (in the standard metric). So, try to re-create the $\varepsilon$-$\delta$ proof and you are done.

Answer 2

You've basically demonstrated that the identity is continuous iff $\exp(x)$ is continuous and I am pretty confident you're allowed to use this as a standard fact in topology (or metric space theory, real analysis, or whatever this course is) that $\exp(x)$ is a differentiable hence continuous function from $\Bbb R$ to $(0,\infty)$. The problem staters also assume you know about $\exp(x)$ (and its basic properties).