Continuity of generalized mean functions

Continuity of generalized mean functions

I'm studying generalized mean functions, and somewhere I found that a
weighted mean function could be defined as $M: (0,\infty)^n \rightarrow
(0,\infty)$ with the properties:
Fixed Point : $M(1,1,\dotsc,1) = 1$
Homogeneity : $M(\lambda x_1, \dotsc, \lambda x_n) = \lambda M(x_1,
\dotsc, x_n)$
Monotonicity : if $x_i \leq y_i$ for each $i$ then $M(x_1,\dotsc, x_n)
\leq M(y_1, \dotsc, y_n)$
This imply some of the properties expected for mean functions, such
boundedness: $\min\{x_i\} \leq M(x_i) \leq \max\{x_i\}$ and continuity.
Though, I was unable to find a proof for continuity. I'm familiar with
real analysis, but not in $\mathbb{R}^n$.
I tried to find a counterexample of such function, only to fail again. The
non-homogeneous functions that would serve as counterexample fail to be
monotonic or to be positive.