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$F$ 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 $\mathop {\lim }\limits_{\left| x \right| \to 0} F(x) = 0$ ?

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My intuition tells me that that is necessary. If there were a counterexample, you would build a series monotonically decreasing in absolute value $x_n \rightarrow 0$ such that either the limit doesn't exist, or isn't $0$. Either way you could build a smooth curve connecting the dots, and that would contradict the given. Although I haven't the time to formalise or even properly verify that thought. – davin Aug 10 '12 at 19:16
@davin :My thought is the same, but when I construct a smooth curve outside $0$, I cannot guarantee this curve is smooth at $0$. – Hezudao Aug 10 '12 at 21:59
1… – user31373 Aug 11 '12 at 5:00
up vote 4 down vote accepted

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

  1. 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 $x=0$;
  2. 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.

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