(topology)
Examples of continuous functions, with which topological properties such as compactness and therefore extremums can be preserved.
This is important because it enables the search for optimal solutions without requiring a continuity of possibilities, which is required when approaching the problem with calculus thus differentiation, so that in addition to requiring the continuity of functions, continuity of the set on which the function is defined (set of possibilities) is also required to ensure differentiability, which is not necessary, as a non-continuous set can as well achieve optimality, and as long as such properties can be transferred consistently in between possibilities (e.g. production sets/function or consumption bundle) and welfare measures (e.g. profit or utility), answering questions such as how to allocate resource (possibilities) to achieve an optimum social welfare state would become possible.
Examples of continuous functions, with which topological properties such as compactness and therefore extremums can be preserved.
This is important because it enables the search for optimal solutions without requiring a continuity of possibilities, which is required when approaching the problem with calculus thus differentiation, so that in addition to requiring the continuity of functions, continuity of the set on which the function is defined (set of possibilities) is also required to ensure differentiability, which is not necessary, as a non-continuous set can as well achieve optimality, and as long as such properties can be transferred consistently in between possibilities (e.g. production sets/function or consumption bundle) and welfare measures (e.g. profit or utility), answering questions such as how to allocate resource (possibilities) to achieve an optimum social welfare state would become possible.
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