Let $H(\mathbf{x},\mathbf{y}):\mathbb{R}_{+}^{l} \times \mathbb{R}_{+}^{m}$ be continuous, strictly increasing in its first $l$ arguments, strictly decreasing in its last $m$ arguments, with $H(\mathbf{0},\mathbf{0})=0$. Define $\Gamma : \mathbb{R}^{l} \to \mathbb{R}^{m}$ by $\Gamma(\mathbf{x})=\{\mathbf{y}\in\mathbb{R}^{m}:H(\mathbf{x},\mathbf{y})\geq0\}$. Show that if $\Gamma(\mathbf{x})$ is compact-valued, then $\Gamma$ is continuous at $\mathbf{x}$.
Symbols Used:
$\mathbb{R}_{+}^{l}$ is the subspace of $\mathbb{R}^{l}$ containing nonnegative vectors.
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I try to use monotonicity, but still can't prove it.