lower semicontinuous functions

A lower semicontinuous function is a type of mathematical function that, at any point, doesn't suddenly jump downward when approaching from nearby points. In other words, its value at a specific point is always less than or equal to the limit of values approaching that point from nearby locations. This property ensures the function’s "downs" are smooth and predictable, making it useful in optimization and analysis because it guarantees the function’s minimum values can be approached by sequences of points where the function is almost as low as at the minimum.