Is $f(x)=x$ always continuous between $\Bbb R$ with an arbitrary metric and $\Bbb R$ with the euclidean metric? Basically the title.  Let $f$ be the function from $\Bbb R$ under any metric $\rho$ to $\Bbb R$ under the euclidean metric defined by $f(x)=x$.  Is $f$ guaranteed to be continuous?  It feels sensible that if I take $\rho(x, y)$ to be arbitrarily small, I should somehow be able to make $|x-y|$ arbitrarily small, but I'm having trouble coming up with an explicit epsilon-delta argument.  Is there some counterexample I'm not seeing?

Let $g(x)=x$ for $x \neq 0,1$. Let $g(0)=1$ and $g(1)=0$. Let $D(x,y)=|g(x)-g(y)|$. Then $D$ is  metric on $\mathbb R$. If $f$ is  continuous w.r.t this metric then there exists $\delta >0$ such that $|x-y| <\frac 1 2$ whenever $|g(x)-g(y)| <\delta$. Take $x=0$ and $y=1+\frac {\delta} 2$ to get  a contradiction.
