Let $X\subset \mathbb{R}^n$ and $Y\subset \mathbb{R}^m$, and $f:X\times Y\to\mathbb{R}$. $f$ is continuously differentiable, and for each $y\in Y$, $f(\cdot,y)$ is strictly convex. Consider the optimization problem $$ \min_x f(x,y) $$ subject to the set of constraints $$ x_i\geq 0 $$ for all $i$. Because of the strict convexity, this optimization problem has a unique solution for each $y.$
Denote by $I(y)=\{i:\lambda_i>0\}$, where $\lambda_i$ is the Lagrange multiplier associated with the constraint $x_i\geq 0$. In other words, $I(y)$ is the set of binding constraints for a given $y$.
Denote by $x(y)$ the solution to the optimization problem under a given $y$, and consider the derivative $\frac{dx}{dy}$ evaluated at some point $y^0$. Is it true that for each $i\in I(y)$, $\frac{dx_i}{dy}=0$? That is, that a marginal change in $y$ does not affect the set of binding constraints? This seems very intuitive but I struggle to write a proof. Any help would be much appreciated!