Uniform continuity is a mathematical concept that describes the behavior of a function as its input values approach each other. It means that for any two points in the function's domain that are close together, the corresponding output values will also be close together.
An example of a function that is not uniformly continuous is the product of two functions, f(x) and g(x), where f(x) is continuous but g(x) is not. This means that as the input values get closer together, the output values may not necessarily get closer together.
If the function g(x) has a discontinuity at a point in its domain where f(x) is continuous, then the product f*g will also have a discontinuity at that point. This means that the function will not meet the criteria for uniform continuity.
The lack of uniform continuity in a function can have implications for its behavior and properties. For example, it may indicate that the function has discontinuities or oscillations that make it difficult to analyze or use in certain mathematical contexts.
The uniform continuity of a function can be determined by analyzing its behavior at different points in its domain, including points where it may have discontinuities. It can also be proven or disproven using mathematical techniques such as the epsilon-delta definition of continuity.