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The definition states that a Utility function represents the preference relation because the relation on R satisfies transitivity and completeness. Yet, strict preferences (and indifference preferences) are not complete relations.
How can they be represented by a Utility function?
Intuitively, representation by a utility function is just assigning numbers to things so that the order of these numbers would tell us exactly the order of preference. This can be defined for weak or strict preferences. For strict preferences, this can be written as $x \succ y$ if and only if $U(x) > U(y)$. You could also say $x \sim y$ if and only if $U(x) = U(y)$ for indifference. Clearly such $U$ exist.
If you are asking when this is possible then the axioms for the strict preferences are different from weak preferences. In fact, they have to be since as you correctly noted these preferences are not complete.
For finite choice spaces, strict preferences are representable by a utility function, if and only if the strict relation is asymmetric and negatively transitive. That is -
Negatively transitive: If $x \succ y$, then for any $z$, either $x \succ z$, or $z \succ y$, or both.
Asymmetric: For no pair $x$ and $y$, $x \succ y$ and $y \succ x$.
If the weak preference $\succeq$ is complete and transitive then the strict preference $\succ$ is asymmetric and negatively transitive. So these conditions are two sides of the same coin - you can build utility representations from strict preferences and indifferences or from weak preferences, they will be equivalent.
For infinite spaces (since you mention $\mathbb{R}$), it is a little more work, and you would also need continuity.
