Strict preference relations and utility representations Suppose I have a rational preference relation $\succsim$ on some consumption set $X$. 

Suppose also that there is a utility function $u:X \to \mathbb{R}$ representing $\succsim$. 

Definition: A function $u: X \to \mathbb{R}$ is a utility function representing preference relation $\succsim$ if, for all $x, y  \in X$, $$x \succsim y \iff u(x) \geq u(y)$$

Is it possible to prove that $x \succ y \iff u(x) &gt; u(y)$ without a continuity condition on $\succsim$?

My intuition says no, but am having difficult finding a suitable counter example. Any help is appreciated.
Yes it is: 
If direction
$$
x \succ y \Rightarrow x \not \precsim y \Rightarrow u(x) &gt; u(y).
$$
Only if direction: 
For all $x, y  \in X$, 
$$
x \succsim y \iff u(x) \geq u(y)
$$
implies
$$
x \sim y \iff u(x) = u(y).
$$
Also
$$
u(x) &gt; u(y) \Rightarrow u(x) \geq u(y) \Rightarrow  x \succsim y ,
$$
$$
u(x) &gt; u(y) \Rightarrow u(x) \not = u(y) \Rightarrow  x \not\sim y.
$$
and
$$
x \succsim y \mbox{ AND } x \not\sim y \Rightarrow x \succ y.
$$