🏃🏽‍♀️‍➡️Intro to Mathematical Analysis Unit 7 – Uniform Continuity

Definition and Concept

• Uniform continuity is a stronger form of continuity that applies to functions defined on a metric space
• A function $f: X \to Y$ is uniformly continuous if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x_1, x_2 \in X$, if $d_X(x_1, x_2) < \delta$, then $d_Y(f(x_1), f(x_2)) < \varepsilon$
  • The choice of $\delta$ depends only on $\varepsilon$ and not on the specific points $x_1$ and $x_2$
• Intuitively, uniform continuity ensures that the function has a consistent rate of change throughout its domain
• The concept is particularly useful when dealing with functions defined on unbounded domains or when studying the behavior of functions at infinity

Comparison with Regular Continuity

• Regular continuity is a local property, meaning that a function can be continuous at a point without being continuous on its entire domain
  • A function $f: X \to Y$ is continuous at a point $a \in X$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x \in X$, if $d_X(x, a) < \delta$, then $d_Y(f(x), f(a)) < \varepsilon$
• In contrast, uniform continuity is a global property that requires the function to have a consistent rate of change throughout its domain
• Every uniformly continuous function is continuous, but not every continuous function is uniformly continuous
  • For example, $f(x) = x^2$ is continuous on $\mathbb{R}$ but not uniformly continuous
• Uniform continuity is a stronger condition than regular continuity and is often required when working with sequences of functions or when studying the behavior of functions at infinity

Key Properties

• Uniform continuity is preserved under composition: if $f: X \to Y$ and $g: Y \to Z$ are uniformly continuous, then $g \circ f: X \to Z$ is also uniformly continuous
• Uniform continuity is preserved under addition and multiplication: if $f, g: X \to \mathbb{R}$ are uniformly continuous, then $f + g$ and $f \cdot g$ are also uniformly continuous
• If a function is uniformly continuous on a compact metric space, then it is also continuous on that space
  • This is because compact sets have the property that every open cover has a finite subcover, which allows for the construction of a suitable $\delta$ for uniform continuity
• Uniform continuity is a topological property, meaning that it is preserved under homeomorphisms between metric spaces
• If a function is Lipschitz continuous, then it is also uniformly continuous
  • A function $f: X \to Y$ is Lipschitz continuous if there exists a constant $L > 0$ such that for all $x_1, x_2 \in X$, $d_Y(f(x_1), f(x_2)) \leq L \cdot d_X(x_1, x_2)$

Epsilon-Delta Definition

• The epsilon-delta definition of uniform continuity is a precise mathematical formulation of the concept
• A function $f: X \to Y$ is uniformly continuous if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x_1, x_2 \in X$, if $d_X(x_1, x_2) < \delta$, then $d_Y(f(x_1), f(x_2)) < \varepsilon$
  • The key difference between this definition and the definition of regular continuity is that the choice of $\delta$ depends only on $\varepsilon$ and not on the specific points $x_1$ and $x_2$
• To prove that a function is uniformly continuous using the epsilon-delta definition, one must find a suitable $\delta$ for any given $\varepsilon > 0$ and show that the inequality holds for all pairs of points in the domain
• The epsilon-delta definition is often used in conjunction with other properties of uniformly continuous functions, such as the preservation of uniform continuity under composition and the relationship between uniform continuity and compactness

Examples and Counterexamples

• Example of a uniformly continuous function: $f(x) = \sin(x)$ on $\mathbb{R}$
  • For any $\varepsilon > 0$, choose $\delta = \varepsilon$. Then, for all $x_1, x_2 \in \mathbb{R}$, if $|x_1 - x_2| < \delta$, then $|\sin(x_1) - \sin(x_2)| < \varepsilon$ (using the mean value theorem)
• Example of a continuous but not uniformly continuous function: $f(x) = x^2$ on $\mathbb{R}$
  • For any $\delta > 0$, there exist points $x_1, x_2 \in \mathbb{R}$ such that $|x_1 - x_2| < \delta$ but $|x_1^2 - x_2^2| > \varepsilon$ for some $\varepsilon > 0$
• Example of a uniformly continuous function on a compact set: any continuous function $f: [a, b] \to \mathbb{R}$
  • By the Heine-Cantor theorem, every continuous function on a compact set is uniformly continuous
• Counterexample of a discontinuous function that is uniformly continuous: the constant function $f(x) = c$ for any $c \in \mathbb{R}$
  • Although the function is not continuous at any point (unless the domain is a singleton), it is uniformly continuous since for any $\varepsilon > 0$, we can choose $\delta = 1$ and the inequality will always hold

Applications in Analysis

• Uniform continuity is a crucial concept in the study of sequences and series of functions
  • If a sequence of uniformly continuous functions converges pointwise to a function, then the limit function is also uniformly continuous
  • This property is essential in the study of functional analysis and the theory of Banach spaces
• Uniform continuity is used in the proof of the Stone-Weierstrass theorem, which states that any continuous function on a compact Hausdorff space can be uniformly approximated by polynomial functions
• In the study of metric spaces, uniform continuity is used to define the concept of uniform convergence, which is a stronger form of convergence than pointwise convergence
  • A sequence of functions $(f_n)$ converges uniformly to a function $f$ on a set $X$ if for every $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$ and all $x \in X$, $d(f_n(x), f(x)) < \varepsilon$
• Uniform continuity plays a role in the study of fixed point theorems, such as the Banach fixed-point theorem, which guarantees the existence and uniqueness of fixed points for certain types of functions on complete metric spaces

Proofs and Theorems

• The Heine-Cantor theorem states that every continuous function on a compact metric space is uniformly continuous
  • The proof relies on the fact that compact sets have the property that every open cover has a finite subcover
• The composition of uniformly continuous functions is uniformly continuous
  • To prove this, let $f: X \to Y$ and $g: Y \to Z$ be uniformly continuous functions. For any $\varepsilon > 0$, choose $\delta_1 > 0$ such that for all $y_1, y_2 \in Y$, if $d_Y(y_1, y_2) < \delta_1$, then $d_Z(g(y_1), g(y_2)) < \varepsilon$. Then, choose $\delta_2 > 0$ such that for all $x_1, x_2 \in X$, if $d_X(x_1, x_2) < \delta_2$, then $d_Y(f(x_1), f(x_2)) < \delta_1$. Setting $\delta = \delta_2$, we have that for all $x_1, x_2 \in X$, if $d_X(x_1, x_2) < \delta$, then $d_Z(g(f(x_1)), g(f(x_2))) < \varepsilon$, proving that $g \circ f$ is uniformly continuous
• The sum and product of uniformly continuous functions are uniformly continuous
  • The proofs follow a similar structure to the composition proof, using the triangle inequality and the properties of the absolute value function
• If a function is Lipschitz continuous, then it is uniformly continuous
  • To prove this, let $f: X \to Y$ be a Lipschitz continuous function with Lipschitz constant $L > 0$. For any $\varepsilon > 0$, choose $\delta = \frac{\varepsilon}{L}$. Then, for all $x_1, x_2 \in X$, if $d_X(x_1, x_2) < \delta$, we have $d_Y(f(x_1), f(x_2)) \leq L \cdot d_X(x_1, x_2) < L \cdot \frac{\varepsilon}{L} = \varepsilon$, proving that $f$ is uniformly continuous

Common Pitfalls and Misconceptions

• A common misconception is that every continuous function is uniformly continuous
  • This is not true, as demonstrated by the example $f(x) = x^2$ on $\mathbb{R}$, which is continuous but not uniformly continuous
• Another misconception is that uniform continuity is a local property
  • Uniform continuity is a global property that requires the function to have a consistent rate of change throughout its entire domain
• Some students may confuse the definitions of uniform continuity and Lipschitz continuity
  • While every Lipschitz continuous function is uniformly continuous, the converse is not true
  • For example, $f(x) = \sqrt{x}$ on $[0, \infty)$ is uniformly continuous but not Lipschitz continuous at $x = 0$
• It is important to remember that the choice of $\delta$ in the epsilon-delta definition of uniform continuity depends only on $\varepsilon$ and not on the specific points in the domain
  • This is a key difference between uniform continuity and regular continuity
• When proving that a function is not uniformly continuous, it is essential to demonstrate that for some $\varepsilon > 0$, no suitable $\delta$ exists that satisfies the epsilon-delta definition for all pairs of points in the domain
  • A common mistake is to show that the function is not uniformly continuous at a specific point, which is not sufficient to prove that the function is not uniformly continuous on its entire domain