If a utility function represents a consumers preference relation if it assigns 'higher numbers' to preferred bundles, how do we know the properties of this preference relation (i.e. complete, transitive, continuous, strictly monotonic?)
For instance, if i have a utility function:
$$U(x_{1}, x_{2}) = |x_{2} - x_{1}|$$
It is clearly continuous, but how can i see whether it's transitive, complete, strictly monotonic etc? That is, how do we construct a unique preference relation from a utility function?
It helps first to generalize your utility function. Let $\ U:\mathbb{R}^2\rightarrow \mathbb{R} \ $ such that $(x_1,x_2)\rightarrow U(x_1,x_2)\equiv|x_2-x_1| \ $
Let $ \ \succsim \ \subseteq X\equiv \mathbb{R}^2 \times \mathbb{R}^2$. By definition, $ \ \succsim \ $ is represented by $\ U \ $ if and only if $ \ \forall a_1,a_2 \in X, \ a_1 \succsim a_2 \Leftrightarrow U(a_1) \geq U(a_2) \ $.
Now you are set up to determine if the preference relation satisfies the aforementioned properties.
For example, $ \ \succsim \ $ is complete if and only if $ \ \forall a_1,a_2\in X, a_1 \succsim a_2 \ or \ a_2 \succsim a_1 \ $. Using the representative utility function, that means $ \ \forall (x_1,x_2),(y_1,y_2)\in \mathbb{R}^2, U(x_1,x_2)\geq U(y_1,y_2) \ $ or $ \ U(y_1,y_2)\geq U(x_1,x_2) $. This holds true by the completeness property of real numbers.
The transitivity argument holds similarly.
The preference relation represented by $ \ U \ $, however, is not strictly monotonic. Note that $ \succsim \ $ is strictly monotonic if and only if $ \ \forall a_1,a_2 \in X, \ a_1 \gneq a_2 \Rightarrow a_1 \succ a_2 $. Suppose $ \ a_1 \equiv (1,0) \ $ and $ \ a_2 \equiv (-1,-2) $. Then $ a_1 \gneq a_2 \ $ but $ \ U(1,0) < U(-1,-2) \ $.
In summary, if the utility function represents a preference relation, you can use properties of that function to determine properties of that preference relation.