Upper semicontinuity refers to a property of set-valued mappings where, intuitively, the image of a point under the mapping does not jump upward as the input approaches that point. This means that for any point in the domain, if you look at nearby points, the values in the image either stay the same or decrease, making it easier to analyze convergence and stability in various mathematical contexts. This concept plays a critical role in understanding the behavior of multifunctions and their differentiability, as well as establishing existence results for equilibrium problems.
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