Uniform Continuity Definition: A function f is said to be uniformly continuous on the interval [a, b] if the following two conditions hold true: 1. Co...
Uniform Continuity Definition:
A function f is said to be uniformly continuous on the interval [a, b] if the following two conditions hold true:
Continuity at every point in (a, b): For every point c in (a, b), the limit of f(x) as x approaches c is equal to f(c).
Continuity from the left and right: The function's limit from the left and right sides of every point in (a, b) must exist and be equal.
In other words, uniform continuity means that the function behaves smoothly and has the same limit from both the left and right sides at every point within the interval.
Examples:
A function like f(x) = 1/x is uniformly continuous on (0, ∞) because it is continuous everywhere and has the same limit from the left and right sides of every point.
A function like f(x) = x^2 is not uniformly continuous on (0, ∞) because its limit from the left and right sides is not equal at x = 0.
A function like f(x) = sin(x) is not uniformly continuous on (0, ∞) because its limit from the left and right sides is not defined at x = 0