Strict preference relations and utility representations

Question

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 $x \succ y \iff u(x) > u(y)$ without a continuity condition on $\succsim$?

My intuition says no, but am having difficult finding a suitable counter example. Any help is appreciated.

Answer 1

Yes it is:
If direction $$ x \succ y \Rightarrow x \not \precsim y \Rightarrow u(x) > u(y). $$ 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 $$ u(x) > u(y) \Rightarrow u(x) \geq u(y) \Rightarrow x \succsim y , $$ $$ u(x) > u(y) \Rightarrow u(x) \not = u(y) \Rightarrow x \not\sim y. $$ and $$ x \succsim y \mbox{ AND } x \not\sim y \Rightarrow x \succ y. $$