7.2 Comparison with Pointwise Continuity

Uniform continuity is like the superhero version of continuity. It's stronger and more reliable than its pointwise counterpart. While pointwise continuity checks each point individually, uniform continuity ensures smooth behavior across the entire domain.

Understanding the difference is crucial. Uniform continuity guarantees consistent behavior, making it essential for various mathematical applications. It's like having a dependable friend who's always there, no matter where you look in the function's domain.

Uniform vs Pointwise Continuity

Definitions and Properties

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• Pointwise continuity means that for each point in the domain, the function is continuous at that point
  • Pointwise continuity is a local property
• Uniform continuity is a stronger condition that requires the same $\delta$ to work for all points in the domain simultaneously for a given $\varepsilon$
  • Uniform continuity is a global property
• Pointwise continuity does not imply uniform continuity, but uniform continuity always implies pointwise continuity
• Uniform continuity is a more stringent condition than pointwise continuity
  • It requires a function to be continuous at every point and the continuity to be "uniform" across the entire domain
• For a function to be uniformly continuous, the rate at which the function changes must be bounded across the entire domain

Conditions for Uniform Continuity

• If a function is pointwise continuous on a closed and bounded interval $[a, b]$, then it is uniformly continuous on $[a, b]$ (Heine-Cantor Theorem)
• If a function is pointwise continuous and has a bounded derivative on an interval $I$, then it is uniformly continuous on $I$
• If a function is Lipschitz continuous on a domain $D$, then it is uniformly continuous on $D$
  • Lipschitz continuity means there exists a constant $K$ such that $|f(x) - f(y)| \leq K|x - y|$ for all $x, y$ in $D$
• If a function is pointwise continuous on a compact metric space, then it is uniformly continuous (a generalization of the Heine-Cantor Theorem)

Examples of Pointwise Continuity

Functions Pointwise Continuous but Not Uniformly Continuous

• The function $f(x) = 1/x$ on the interval $(0, 1]$ is pointwise continuous but not uniformly continuous
• The function $f(x) = x^2$ on the real line is pointwise continuous but not uniformly continuous
• The function $f(x) = \sin(1/x)$ on the interval $(0, 1]$ is pointwise continuous but not uniformly continuous due to its oscillatory behavior near $0$
• In general, functions with unbounded derivatives or functions with oscillations of increasing frequency near a point are often pointwise continuous but not uniformly continuous

Visualizing Pointwise Continuity

• Consider the function $f(x) = 1/x$ on the interval $(0, 1]$
  • For any point $x_0$ in $(0, 1]$, we can find a small neighborhood around $x_0$ where the function is continuous
  • However, as we approach $0$ from the right, the function values become arbitrarily large, making it impossible to find a single $\delta$ that works for all points in the domain
• The function $f(x) = \sin(1/x)$ on $(0, 1]$ is another example of pointwise continuity without uniform continuity
  • The function oscillates more rapidly as $x$ approaches $0$, making it impossible to find a uniform $\delta$ for a given $\varepsilon$

Uniform Continuity Implies Pointwise Continuity

Proof

• Assume $f$ is uniformly continuous on a domain $D$
  • This means for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x, y$ in $D$, if $|x - y| < \delta$, then $|f(x) - f(y)| < \varepsilon$
• Let $x_0$ be any point in $D$
  • To show pointwise continuity at $x_0$, we need to prove that for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x$ in $D$, if $|x - x_0| < \delta$, then $|f(x) - f(x_0)| < \varepsilon$
• Given $\varepsilon > 0$, by uniform continuity, there exists a $\delta > 0$ such that for all $x, y$ in $D$, if $|x - y| < \delta$, then $|f(x) - f(y)| < \varepsilon$
• In particular, for any $x$ in $D$ with $|x - x_0| < \delta$, we have $|f(x) - f(x_0)| < \varepsilon$
• Thus, $f$ is pointwise continuous at $x_0$
  • Since $x_0$ was arbitrary, $f$ is pointwise continuous on $D$

Intuition

• Uniform continuity is a stronger condition than pointwise continuity
• If a function is uniformly continuous, it means that the function is continuous at every point and the continuity is "uniform" across the entire domain
  • The same $\delta$ works for all points in the domain for a given $\varepsilon$
• Pointwise continuity only requires the function to be continuous at each individual point, without any uniformity condition
• Therefore, if a function is uniformly continuous, it must also be pointwise continuous, but the converse is not necessarily true

Pointwise vs Uniform Continuity on Compact Sets

Heine-Cantor Theorem

• If a function is pointwise continuous on a closed and bounded interval $[a, b]$, then it is uniformly continuous on $[a, b]$
• This theorem establishes a connection between pointwise and uniform continuity on compact sets in $\mathbb{R}$
• The compactness of the interval $[a, b]$ allows us to extend the local property of pointwise continuity to the global property of uniform continuity

Generalization to Compact Metric Spaces

• The Heine-Cantor Theorem can be generalized to compact metric spaces
• If a function is pointwise continuous on a compact metric space, then it is uniformly continuous
• This generalization highlights the role of compactness in bridging the gap between pointwise and uniform continuity

Intuition and Examples

• Compact sets have the property that any open cover of the set has a finite subcover
  • This property allows us to extend local continuity to global continuity
• Consider the function $f(x) = x^2$ on the interval $[0, 1]$
  • The function is pointwise continuous on $[0, 1]$
  • By the Heine-Cantor Theorem, $f(x) = x^2$ is also uniformly continuous on $[0, 1]$
• In contrast, $f(x) = x^2$ is not uniformly continuous on the entire real line, which is not compact
  • This example illustrates the importance of compactness in relating pointwise and uniform continuity