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Proof of the tangency condition in UMP

The proof below is not very rigorous - I do not discuss discontinuities - but I think it captures the logic well. From Wikipedia: Suppose $f$ is a function of one real variable defined on an ...
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Convexity of preferences (dissimilar definitions)

For Q 1: Let me give you a preference relation on $\mathbb{R}^2_+$ $(x_1, y_1) \succsim (x_2, y_2)$ if and only if $(x_1^2 + y_1^2 > x_2^2 + y_2^2)$ or $(x_1^2 + y_1^2 = x_2^2 + y_2^2 \ \wedge x_1 \...
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