The continuity of multivariable function
$F$ is a function on $\mathbb R^n$ such that for every smooth curve $\gamma:[0,1] \rightarrow \mathbb R^n, \gamma(0)=0 $, we have $\mathop {\lim }\limits_{t \to 0} F(\gamma (t)) = 0$, is it necessary that $\mathop {\lim }\limits_{\left| x \right| \to 0} F(x) = 0$ ?
Good question!
For $n=2$, you may find a positive answer to this question in Remarks on the continuity of functions of two variables, by Michael McAsey and Libin Mou. The proof is relatively difficult.
In the paper, the following theorem is proven:
Let $D$ be an open subset of $\mathbf R^2$ containing $(0,0)$, and $f$ a real-valued function on $D$. If $f$ is discontinuous at $(0,0)$, then
- There exists a continuously differentiable convex function $z \in C^1$ such that either $f(x,z(x))$ or $f(z(x),x)$ is discontinuous at $x=0$;
- For any integer $m\geq 0$, there exist functions $x(t), y(t) \in C^m$ such that $f(x(t),y(t))$ is discontinuous at $t=0$.
The authors state a similar theorem for $n=3$ (which also implies a positive answer to your question for $n=3$) and indicate how these results can be further generalized.