Usually the continuity of multi-variate function is defined through something like "$\delta$-disk."
Can we define such continuity with the similar one-dimension definition? As follow: $f:\mathbb R^2\to\mathbb R$ is continuous at $x$ iff $\forall\delta>0\exists\epsilon>0\forall y\in\mathbb R^2$ we have $|x-y|<\epsilon$ implies $|f(y)-f(x)|<\delta$.
If this definition works, then why do we need to complicate the problem? If not, could you please provide a counterexample?
Similarly, there are many very complicated generalization of absolute continuity in $\mathbb R^n$. Why don't people just use the same definition in $\mathbb R^n$ as if in $\mathbb R$ if that works? If it doesn't work in $\mathbb R^n$, then could you please bring a counterexample?
1d definition": $∀ε>0 ∃δ>0$ st $∑_j(y_j−x_j)<δ⟹∑_j|f(x_j)−f(y_j)|<ε$, where each $(x_j,y_j)$ is a subinterval of the domain of f. Can it be directly used in $\mathbb R^n$?