Uniform continuity is a type of continuity in which the rate of change of a function is consistent throughout its entire domain. In other words, the function does not have sudden jumps or breaks in its behavior.
While regular continuity only requires the function to have a consistent rate of change at each point in its domain, uniform continuity requires the function to have a consistent rate of change across the entire domain.
Uniform continuity is important in many areas of science and engineering, as it helps us predict and analyze the behavior of functions in a more consistent and predictable manner. It also allows us to make more accurate calculations and models.
Polynomials, such as x^2, are examples of functions that exhibit uniform continuity. Other examples include trigonometric functions, exponential functions, and rational functions.
To prove uniform continuity, one must show that for any given epsilon (ε), there exists a delta (δ) such that for all x and y in the domain of the function, if the distance between x and y is less than delta, then the difference between the function values at x and y is less than epsilon.