Uniform continuity for two functions f/g means that for every epsilon > 0, there exists a delta > 0 such that for all x,y in the domain of f and g, if the distance between x and y is less than delta, then the distance between f(x) and g(x) is less than epsilon.
Uniform continuity requires that the same delta works for all points in the domain, while regular continuity only requires that a delta exists for each point.
Some common techniques for proving uniform continuity include the use of the mean value theorem, the Cauchy criterion, and the use of Lipschitz functions.
Yes, a function can be uniformly continuous on one interval but not on another. Uniform continuity depends on the behavior of the function in a specific interval, so it is possible for a function to have different uniform continuity on different intervals.
Uniform continuity is closely related to the concept of a limit. Both concepts involve the behavior of a function as the input approaches a certain value. Uniform continuity guarantees that the function will not have any sudden jumps or breaks, which is similar to the idea of a limit being the value that a function approaches as the input gets closer and closer to a certain value.