Point of tangency is an optimal point for a monotone, quasi-concave function

Question

Given $U : \mathbb{R^2} \to \mathbb{R}$ is monotone and quasi-concave, consider the following problem :

$$\max_{(x,y) \ \in \ \mathbb{R}^2}[U(x,y)] \text{ subject to } p_1 x + p_2 y \leq M ; \ (p_1, p_2 > 0)$$

1. If a level curve of $U$ is tangent to $p_1 x + p_2 y = M$ at some point $(a,b)$ on such that the slope of the tangent is $-\frac{p_1}{p_2}$, then show that it is an optimal point.

I have seen the tangency condition to be written as $\displaystyle \frac{df(x)}{dx} = -\frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} = -\frac{p_1}{p_2}$ where $f: (a-\epsilon, a+\epsilon) \to \mathbb{R}$ describes the level curve around the point $(a,b)$.

2. Is it possible to have $\frac{df(x)}{dx} = -\frac{p_1}{p_2}$ but one or both of $\frac{\partial U(x,y)}{\partial x}$ and $\frac{\partial U(x,y)}{\partial y}$ are not defined?

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My attempt:

If $U$ is differentiable at all points, then the Karush-Kuhn-Tucker condition is satisfied and (1) follows. For (2), each term is defined due to our assumption.

The problem lies when $U$ is not differentiable everywhere. In that case, how do we show that the point of tangency is an optimal point? I don't know if the tangency condition requires all the derivatives in (2) to be defined in the interval around $(a,b)$; assume differentiability of $U$ in that particular region around $(a,b)$ only if that is a necessity.

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Definitions:

1. Quasi-concavity of $U$ :

   • (Definition $1$) If $U(r) \geq U(s)$ for $r,s \in \mathbb{R}^2$, then $U(\lambda r + (1-\lambda)s) \geq U(s) \ \forall \lambda \in [0,1]$.

   • (Definition $2$) The set $S_\alpha := \{ (x,y) : U(x,y) \geq \alpha\}$ is convex for all values of $\alpha$.

2. Monotonicity of $U$ :

   • $r \geq s \implies U(r) \geq U(s)$ where $r \geq s$ means $(r_1 \geq r_2) \land (s_1 \geq s_2)$ for $r = (r_1, r_2)$ and $s = (s_1, s_2)$.

Answer 1

For your question 2: there can be a function with discontinuities, which still has smooth level curves.

E.g. $U(x, y) = x + y + \lfloor x+y \rfloor$. It is monotone, quasi-concave, and level curves are straight lines; but the function is discontinuous at lines where $x + y$ is an integer.

For question 1: instead of "tangent", it seems the following lighter condition may work: if a level curve of $U$ has an intersection with $p_1 x + p_2 y = M$, and the level curve is entirely on one side of $p_1 x + p_2 y = M$. That would free us from differentiability. But I have no proof of it.