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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.

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Seems like what you want to do is find the formal math definitions and apply them.

upper contour sets are open

Find a sequence $(x_n,y_n)$ where all elements are in an upper contour set but the limit of the sequence is not.

increase in one component leads to a higher point of utility

Show this, i.e. $$ x' > x \Rightarrow U(x',y) > U(x,y) $$ and $$ y' > y \Rightarrow U(x,y') > U(x,y). $$

combination of two is strictly preferred

Compared to what...? Maybe you are thinking of this definition:
for all $(x,y) \sim (x',y')$ and $\lambda \in (0,1)$ $$ (x,y) \prec (\lambda x + (1-\lambda x'), \lambda y + (1-\lambda y')). $$


Perhaps you mean "how do I show these". Well, this depends on the exact utility function, there is no general solution.
For the lexicograpic preference the first two are properties are not hard if you think about them. The third one is tricky, but the indifference curves for lexicographic preferences are singletons, i.e. there are no two points where $(x,y) \sim (x',y')$, hence the statement is true by default.

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