Non-continuous addition in metric space Is there an example of a metric d on some linear space X, which makes the addition of vectors a non-continuous operation? Where we endow the product space with the metric $d((x, y), (x', y')) = d(x, x') + d(y, y')$.
I know that I can find such a metric for multiplication by a scalar to a be a non-continuous operation, but I'm not sure of an example vector addition. Thanks!

Consider the usual vector space structure on $\mathbb R$. Let $f: \mathbb R \to \mathbb R$ be any bijection such that $ f(\frac 1 n )=\frac 1  n, f(-\frac 1 n )=-\frac 1  n$ but $f(0) \neq 0$. Define $d(x,y)=|f(x)-f(y)|$. Then $d$ is a metric and $f(0)=f(\frac 1 n +(-\frac 1 n)) $ which does not tend to $\lim f(\frac  1 n)+\lim f(-\frac 1 n)$.
