I want to define the local maximum of multivatriable funcrtion. I did not find any formal definition for that, except for two-variable functions.
Is the following definition correct, generalized from the definition of two-variable function?
A function $f(x_1,\cdots,x_n)$ has a $\textit{local maximum}$ at the point $(q_0,\cdots,q_n)$ if $f(q_0,\cdots,q_n) \geq f(q_0+\delta_0,\cdots,q_n+\delta_n)$, for all sufficiently small $\pm \delta_i \in \{\delta_0,\cdots,\delta_n\}$.