In the study of mathematical analysis, continuity is a cornerstone concept that helps us understand the behavior of functions. While the standard notion of continuity is often sufficient for basic applications, there exists a stronger version known as uniform continuity.

Heinrich Heine introduced the concept of uniform continuity in 1872 when he published a proof of the theorem now known
as Heine's theorem for continuous functions on a segment. However, the idea and proof originally came from Dirichlet in 1854.

To lay the foundation, let’s first revisit the classic definition of continuity, or pointwise continuity. A function f
is said to be pointwise continuous at a point x₀ if, for any ε > 0 (epsilon), there exists a δ > 0 (delta) such that for all x within a certain range around x₀:

|x - x₀| < δ implies |f(x) - f(x₀)| < ε.

In simple terms, a continuous function means that small changes in the input x lead to small changes in the output
f(x). However, this condition only needs to hold locally around each point x₀, and the choice of δ can vary depending on both x₀ and ε.

Uniform continuity takes this idea a step further by demanding a stronger condition. A function f is uniformly
continuous on an interval A if, for any ε > 0, there exists a single δ > 0 such that for all pairs of points x and x' in A:

|x - x'| < δ implies |f(x) - f(x')| < ε.

The key difference here is that δ does not depend on the specific points x and x' chosen; it depends only on ε. This
means that once you fix an ε, the same δ works uniformly over the entire interval A, rather than being different at each point.

For example, the functions sin(x) and cos(x) are uniformly continuous. Since their rate of change is bounded (their
derivatives are at most 1 in absolute value), the oscillations of these functions are controlled, allowing for a consistent choice of δ.

There are functions that are continuous but not uniformly continuous. For instance:

The function f(x) = 1/x on the interval (0, 1] is continuous but not uniformly continuous. As x approaches 0, the
function f(x) increases without bound, meaning that no single δ works for all pairs of points within the interval.

The function f(x) = x² on the interval ℝ (all real numbers) is continuous everywhere but not uniformly continuous on
the entire real line. As x becomes very large, small changes in x lead to increasingly large changes in f(x). This prevents a uniform δ from existing for a given ε.

Uniform continuity helps ensure that certain integrals converge and that solutions to differential equations behave
predictably over their domains. When approximating functions using polynomials, uniform continuity also helps guarantee that the approximation error can be controlled across the entire
interval.

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