Question

Prove that if \(f: S \rightarrow T\) is uniformly continuous on \(B \subseteq S,\) and \(g: T \rightarrow U\) is uniformly continuous on \(f[B]\), then the composite function \(g \circ f\) is uniformly continuous on \(B\).

Short answer

The composite \(g \circ f\) is uniformly continuous on \(B\).

Step-by-step solution

Define Uniform Continuity

A function \(h: A \rightarrow B\) is uniformly continuous if, for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(x, y \in A\), if \(||x - y|| < \delta\), then \(||h(x) - h(y)|| < \epsilon\).

Apply Uniform Continuity to f

Since \(f: S \rightarrow T\) is uniformly continuous on \(B \subseteq S\), given \(\epsilon > 0\), there exists a \(\delta_1 > 0\) so that for all \(x, y \in B\), if \(||x - y|| < \delta_1\) then \(||f(x) - f(y)|| < \frac{\epsilon}{2}\).

Apply Uniform Continuity to g

Since \(g: T \rightarrow U\) is uniformly continuous on \(f[B]\), given \(\epsilon > 0\), there exists a \(\delta_2 > 0\) so that for all \(u, v \in f[B]\), if \(||u - v|| < \delta_2\) then \(||g(u) - g(v)|| < \epsilon\).

Combine the Conditions for g and f

Utilize the \(\delta_1\) from Step 2 and the \(\delta_2\) from Step 3. If for \(x, y \in B\), \(||x - y|| < \delta_3\) where \(\delta_3 = \min(\delta_1, \delta_2)\), then \(||f(x) - f(y)|| < \frac{\epsilon}{2}\), and since \(f(x), f(y) \in f[B]\) as \(||f(x) - f(y)|| < \delta_2\), we have \(||g(f(x)) - g(f(y))|| < \epsilon\).

Conclude Uniform Continuity of Composite

Therefore, for \(g \circ f\), given \(\epsilon > 0\), there exists \(\delta_3 > 0\) such that \(||x - y|| < \delta_3\) leads to \(||g(f(x)) - g(f(y))|| < \epsilon\), thus proving \(g \circ f\) is uniformly continuous on \(B\).

Key concepts

Composite Functions

When dealing with functions, especially in advanced mathematics, the idea of composite functions is crucial. A composite function is formed when one function is applied to the results of another. In mathematical terms, if we have two functions, say \(f\) and \(g\), the composite function \(g \circ f\) applies function \(f\) first and then \(g\) on the resulting values.
This can be symbolized as:

• \(f: S \rightarrow T\)
• \(g: T \rightarrow U\)
• \(g \circ f: S \rightarrow U\)

This means that the output of \(f\) becomes the input for \(g\). Understanding composite functions is vital when analyzing continuity and other properties through consecutive transformations, as the behavior of the output depends on both functions acting in tandem.
In the exercise at hand, we are looking at how the uniform continuity of individual functions \(f\) and \(g\) ensures the uniform continuity of the composite \(g \circ f\). This is a foundational concept in mathematical analysis, often serving as a stepping stone to more intricate properties like differentiability and integrals.

Epsilon-Delta Definition

The epsilon-delta definition is a fundamental part of real analysis and calculus, used to rigorously define the notion of continuity. This concept ties closely with both pointwise and uniform continuity. To say that a function \(h: A \rightarrow B\) is uniformly continuous means that the distance between outputs \(h(x)\) and \(h(y)\) can be made arbitrarily small whenever the distance between \(x\) and \(y\) is sufficiently small.
The definition goes like this:

• For every \(\epsilon > 0\) given (which represents how close the function outputs need to be),
• there exists a \(\delta > 0\) (representing how close the inputs need to be) such that for all points \(x, y\) in the domain of \(h\),
• if \(||x - y|| < \delta\), then \(||h(x) - h(y)|| < \epsilon\).

This uniform condition must hold for the entire domain of \(h\), making it stronger than regular continuity where \(\delta\) may depend on the point \(x\) in question. The exercise illustrates applying this principle to prove that a composite function \(g \circ f\) retains uniform continuity when both \(f\) and \(g\) possess this trait.

Uniform Convergence

Uniform convergence is another important concept in analysis, often considered alongside uniform continuity. It's a type of convergence related to sequences or series of functions. The importance of uniform convergence comes into play during the approximation of functions and ensuring properties like integration and differentiation are preserved through the limit process.
For example, if a sequence of functions \(\{f_n(x)\}\) converges to a function \(f(x)\) uniformly on some set, it implies:

• For any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n \geq N\),
• and for all \(x\) in the set, \(||f_n(x) - f(x)|| < \epsilon\).

The distinguishing factor is the independence of \(x\) in choosing \(N\), maintaining the closeness of \(f_n(x)\) to \(f(x)\) uniformly over the entire set. Although uniform convergence is more about sequences rather than single functions being continuous, understanding it provides deeper insight into how the behavior of functions can be controlled and predicted over entire intervals or domains.
In the context of our exercise, uniform continuity and uniform convergence both share the idea of control over entire domains, helping ensure smooth and consistent transformations in functions, especially when analyzing compositional behavior.