Is quasinorm always continuous?

Question

Let X be a vector space over $\mathbb{R}$ or $\mathbb{C}$, $\|\cdot\|: X\rightarrow [0,\infty)$ is called a quasi-norm if

i) $\|x\|=0 \Rightarrow x=0$

ii) $\|\lambda x\|=|\lambda|\|x\|, \forall \lambda, x$

iii) $\exists K\ge 1$, s.t. $\|x+y\|\le K(\|x\|+\|y\|), \forall x,y$

My question is:

If $\|x_k-x\|\rightarrow 0$, can we conclude that $\|x_k\|\rightarrow \|x\|$?

Answer 1

No. A counterexample for $\mathbb R^2$ is: $$\|\langle x,y\rangle\|=\begin{cases} K|x|&\text{for } y=0\\|x|+|y|&\text{otherwise} \end{cases}$$ for some $K>1$.