The $p$-adic norm of an integer $k$ is $||k||_p := p^{-\alpha}$ where $p^\alpha$ is the largest power of $p$ that divides $k$.
This function can be easily extended to the rationals as $||a/b||_p := ||a||_p/||b||_p$ (notice that's well defined).
Apparently there is this Chevalley's Theorem which states that this norm can be even extended to the real numbers preserving several nice proprieties.
Is this extended norm continuous?
It is not.
If it were continuous, it would be continuous at $0$. Then, there would exist $\delta>0$ such that if $|x|<\delta$ then $|||x||_p- ||0||_p|=||x||_p<1$.
However, this is impossible since $||\frac{1}{p^k}||_p = \frac{||1||_p}{||p^k||_p}=\frac{p^{-0}}{p^{-k}}=p^k \to \infty$ and for $k$ big enough we will have that $|\frac{1}{p^k}|<\delta$.