I have a midterm coming up and I am still not entirely sure on the formal arguments for proving (strict)\convexity, monotonicity, continuity, quasi-concavity e.t.c. I think I have a pretty strong grasp on the intuition behind these concepts, but I lack rigorous formal math reasoning. This is one of the times where my minimalistic math background comes to haunt me.
For instance, I know this about lexicographic preferences:
Not continuous because upper contour sets are open.
Strongly monotonic as increase in one component leads to a higher point of utility
Strictly convex because combination of two is strictly preferred.
But how would I put that into formal math? If someone could just work through one example I think I can figure it out once or for all, or guide me to some lecture notes where one particular example is worked out formally.