Uniform continuity is a mathematical concept that describes the behavior of a function. A function is considered uniformly continuous if the amount of change in the output values is proportional to the amount of change in the input values, regardless of where on the function's domain the changes occur.
Uniform continuity is a stronger condition than regular continuity. While regular continuity only requires that the function is continuous at every point in its domain, uniform continuity requires that the function is also continuous at every point in its domain when the domain is extended to include all real numbers.
A function must be continuous at every point in its domain, and its rate of change must not vary significantly between points in its domain. In other words, as the input values get closer to each other, the output values must also get closer together at a consistent rate.
To prove that a function is uniformly continuous, you must show that for any given positive number ε, there exists a positive number δ such that whenever two points in the function's domain are within δ units of each other, their corresponding output values are within ε units of each other. This can be done using the ε-δ definition of uniform continuity.
No, not all continuous functions are uniformly continuous. While all uniformly continuous functions are also continuous, the reverse is not always true. For example, the function f(x) = 1/x is continuous on its domain of (0,∞), but it is not uniformly continuous because its rate of change becomes unbounded as x approaches 0.