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.
Continuity is a mathematical concept that describes the smoothness and connectedness of a function. A function is considered continuous if it has no abrupt changes or breaks in its graph. Uniform continuity is a stronger form of continuity that requires the function to have a consistent rate of change over its entire domain.
A function is continuous if it satisfies the epsilon-delta definition, which states that for any small value of epsilon, there exists a corresponding small value of delta such that when the distance between two points on the graph of the function is less than delta, the difference between the corresponding function values is less than epsilon. Uniform continuity can be determined by examining the function's rate of change and ensuring that it does not have any sudden changes or discontinuities.
Continuity and uniform continuity are important concepts in fields such as physics, engineering, and economics. In physics, these concepts are used to describe the smoothness of motion and to model physical systems. In engineering, continuity is essential for designing smooth and efficient systems, while uniform continuity is crucial for maintaining stability. In economics, continuity is used to analyze the behavior of markets and predict future trends.
Yes, a function can be continuous but not uniformly continuous. This means that the function may have no abrupt changes or breaks, but its rate of change may vary over different parts of its domain. For a function to be uniformly continuous, its rate of change must be consistent over its entire domain.
To prove that a function is uniformly continuous, you can use the epsilon-delta definition or the limit definition of uniform continuity. Additionally, you can also use the intermediate value theorem, which states that if a function is continuous on a closed interval, it must also be uniformly continuous on that interval. Other methods of proof include using the mean value theorem or the Cauchy criterion for uniform continuity.