A weaker definition of local non-satiation can also imply indifference "curve" Let $u$ be a continuous utility function on $\mathbb R^2_+\setminus\{0\}$. Consider the following three conditions:


Local non satiation says that for any $x \in X$ and $\epsilon &gt; 0$, there exists $y \in X$ such that $d(x,y) &lt; \epsilon$ and $U(x) &lt; U(y)$.
Local non satiation* says that for any $x \in X$ and $\epsilon &gt; 0$, there exists $y \in X$ such that $d(x,y) &lt; \epsilon$ and $U(x) \neq U(y)$.
The indifference sets of $U$ are curves.


As a standard result, (1) implies (3) and (3) implies(1). 

Obviously, (1) implies (2) so (3) also implies (2). 

Can (2) also imply (1) and (3)？
(2) does not imply (1). Consider a utility function with "circular indifference curves", e.g. $u(x,y)=-(x-1)^2-(y-1)^2$. At the bliss point $(1,1)$, the function satisfies (2) but violates (1).

(2) does imply (3), and the proof should be similar to the one showing that (1) implies (3).