The discussion revolves around the properties of a function \( f: \mathbb{R} \to \mathbb{R} \) that is increasing on a dense set, and the implications of its continuity and uniform continuity on another function \( g(x) = \inf_{x There is no consensus on the implications of continuity and uniform continuity between the functions \( f \) and \( g \), as the initial question remains open for discussion. Additionally, there is a disagreement regarding the appropriate forum for self-study questions. The discussion does not resolve the mathematical implications of continuity and uniform continuity, nor does it clarify the definitions or assumptions regarding the dense set or the function \( g \).Discussion Character
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student12s said:I´ve been trying to solve this for some time: Let f: R to R be an increasing on a dense set. Define g(x)=inf_{x<t in D} f(t). Show that continuity of f does not imply continuity of g but uniform continuity of f does imply uniform continuity of g.
Any help?
student12s said:I encounter this question while studying Probability by my own. It's not homework/coursework. The application is understanding properties of distribution functions.