I want to know if there is any mistake in my proof. Please bear with me the proof is a bit long a not very pretty. This seems very simple but I cannot think of a better way to prove it.
Let $||\cdot||_p:\Bbb R^n\to \Bbb R$ be defined, for $p\ne 0$, by $$ ||(x_1,\dots,x_n)||_p:=\left( \sum_{i=1}^n |x_i|^p\right)^{1/p}, $$ I'll try to show that, for a fixed $x\in\Bbb R^n$, $M(p):=||x||_p$ is continuous on $\Bbb R\backslash \{0\}$.
Let $f:(0,\infty)\times\Bbb R\to \Bbb R$ be defined by $$ f(x,y):=x^y. $$ Since $ x^y=e^{y \ln x} $, so $$f=\exp\circ\ t\circ g$$ where $t(x,y)= yx$ and $g(x,y)=(\ln x,y)$. Since each of these $3$ functions is continuous, $f$ is continuous.
Let $h:\Bbb R\backslash\{0\}\to\Bbb R\times (\Bbb R\backslash\{0\})$ be defined as $$ h(r):=\left( \sum_{i=1}^n |x_i|^r,1/r \right). $$ We can see that $h$ is continuous also.
Since $M=f\circ h$, $M$ is continuous. QED
Note: I frequently make use of that fact that a function is continuous iff each of its component is continuous. Also, the trivial case where $x_1=\dots=x_n=0$ is not mentioned (I was too lazy to define $0^0$).
PS. If any of you have a shorter proof, I would be more than happy to look at it.