Uniformly Continuous Function

A function is said to be uniformly continuous on the interval [a, b] ⊆ ℝ if for every $ \epsilon > 0 $ there exists a $ \delta = \delta(\epsilon) > 0 $ such that $$ |x - x'| < \delta \quad \Rightarrow \quad |f(x) - f(x')| < \epsilon $$ for all $ x, x' \in [a, b] $.

Geometrically, this means that a continuous function on [a, b] never exhibits any sudden, unbounded oscillations.

uniformly continuous function

Intuitively, smaller values of $ \epsilon $ require correspondingly smaller values of $ \delta $, and vice versa.

In general, for an ordinary continuous function, $ \delta $ might depend on both $ \epsilon $ and the specific point $ x_0 $ at which continuity is evaluated:

$$ \delta = \delta(\epsilon, x_0) $$

In contrast, for uniformly continuous functions, the choice of $ \delta $ depends solely on $ \epsilon $, independently of any particular $ x_0 $.

Note: You can think of the relationship between $ \epsilon $ and $ \delta $ as defining a rectangle centered at any point $ x \in [a, b] $. For a uniformly continuous function, the graph only intersects the vertical sides of this rectangle - it never crosses the horizontal edge. For example, if one can choose $ \delta = 2\epsilon $ on the interval [a, b], the graph meets these rectangles strictly along their vertical boundaries, confirming uniform continuity. Conversely, if on some interval such as [0, a] the function only satisfies $ \delta = \epsilon/2 $ and its graph crosses the bottom side of the rectangle, then uniform continuity fails on that interval.
example

A Practical Example
Difference Between Continuity and Uniform Continuity

A Practical Example

Consider the function $ f(x) = x^3 $, which is continuous on ℝ:

$$ f(x) = x^3 $$

This function is also uniformly continuous on the interval [0, 1].

Fix any $ \epsilon > 0 $. Uniform continuity guarantees the existence of some $ \delta > 0 $ (depending solely on $ \epsilon $) such that for any $ x_1, x_2 \in [0, 1] $:

$$ |x_1 - x_2| < \delta \quad \Rightarrow \quad |x_1^3 - x_2^3| < \epsilon $$

Observing that the difference of cubes factors as:

$$ |x_1^3 - x_2^3| = |x_1 - x_2| \cdot |x_1^2 + x_1x_2 + x_2^2|, $$

and noting that for $ x_1,x_2 \in [0,1] $ we have

$$ 0 \le x_1^2,\, x_1x_2,\, x_2^2 \le 1, $$

so that

$$ |x_1^2 + x_1x_2 + x_2^2| \le 3, $$

it follows that

$$ |x_1^3 - x_2^3| \le 3\, |x_1 - x_2|. $$

Thus, to ensure $ |x_1^3 - x_2^3| < \epsilon $, it suffices to require

$$ 3\,|x_1 - x_2| < \epsilon, $$ which is equivalent to

$$ |x_1 - x_2| < \frac{\epsilon}{3}. $$

In conclusion, for every $ \epsilon > 0 $ one may choose

$$ \delta = \frac{\epsilon}{3}, $$

which verifies that $ f(x) = x^3 $ is uniformly continuous on [0, 1].

Difference Between Continuity and Uniform Continuity

While the two concepts are related, they are not equivalent:

A uniformly continuous function on [a, b] is necessarily continuous, with its $ \delta $ chosen uniformly over the whole interval, i.e., $ \delta = \delta(\epsilon) $ independent of the point.
A continuous function on [a, b] may have a $ \delta $ that depends on both $ \epsilon $ and the point $ x_0 $, i.e., $ \delta = \delta(\epsilon, x_0) $, and hence does not guarantee uniform behavior over the interval.

Thus, uniform continuity is a strictly stronger condition than mere continuity.

Note: In short, while continuity is a necessary condition for uniform continuity, it is not sufficient on its own.

And so on.