Question

A uniformly continuous function of a uniformly continuous function is uniformly continuous. State this more precisely and prove it.

Short answer

Question: Prove that if we have two uniformly continuous functions, the composition of these functions is also uniformly continuous. Answer: To prove the composition of two uniformly continuous functions is also uniformly continuous, we used the definitions of uniform continuity for both functions and followed a series of steps to demonstrate that for every ε > 0, there exists a δ > 0 such that for every x, y in the domain, the difference between their images under the composition function h(x) = g(f(x)) is smaller than ε. This confirms that h(x) is also uniformly continuous.

Step-by-step solution

Definition of Uniformly Continuous Function

A function f: A -> B is said to be uniformly continuous if for every ε > 0, there exists a δ > 0 such that for every x, y in A, if |x - y| < δ, then |f(x) - f(y)| < ε.

Statement to Prove

Let f: A -> B and g: B -> C be two uniformly continuous functions. The objective is to prove that the composition of these functions, h(x) = g(f(x)), is also uniformly continuous.

Assume f and g are Uniformly Continuous

Assume f and g are uniformly continuous functions. This means that for any ε1 > 0, there exists a δ1 > 0 such that for every x, y in A, if |x - y| < δ1, then |f(x) - f(y)| < ε1. Similarly, for any ε2 > 0, there exists a δ2 > 0 such that for every u, v in B, if |u - v| < δ2, then |g(u) - g(v)| < ε2.

Choose ε2

Choose ε2 to be equal to ε > 0, where ε is the desired precision for the composition function h(x) = g(f(x)).

Find δ2

Since g is uniformly continuous, there exists a δ2 > 0 such that for every u, v in B, if |u - v| < δ2, then |g(u) - g(v)| < ε2.

Choose ε1

Choose ε1 to be equal to δ2 from the previous step.

Find δ1

Since f is uniformly continuous, there exists a δ1 > 0 such that for every x, y in A, if |x - y| < δ1, then |f(x) - f(y)| < ε1.

Show that h is Uniformly Continuous

Now, let's show that h(x) = g(f(x)) is uniformly continuous. Let x, y ∈ A such that |x - y| < δ1. Since f is uniformly continuous, we have |f(x) - f(y)| < ε1 = δ2. Now let u = f(x) and v = f(y). Hence, |u - v| < δ2. Since g is uniformly continuous, we have |g(u) - g(v)| < ε2 = ε which implies |g(f(x)) - g(f(y))| < ε. Therefore, h(x)=g(f(x)) is uniformly continuous. This completes the proof that the composition of two uniformly continuous functions is also uniformly continuous.

Key concepts

Composition of Functions

When we discuss the composition of functions, we're talking about combining two or more functions to form a new function. This is done through the process of function composition, which means applying one function to the results of another. If we have two functions, say, \(f\) and \(g\), their composition is denoted as \(h(x) = g(f(x))\). Here, \(f(x)\) is computed first and then \(g\) is applied to \(f(x)\).

There are a few things to keep in mind:

• The composition order matters: \(g(f(x))\) is generally not the same as \(f(g(x))\).
• For the composition to work, the range of \(f\) must fit into the domain of \(g\).

In the context of continuity, the uniform continuity properties of \(f\) and \(g\) make it possible to claim and prove similar properties for their composition, \(h(x)\).

This becomes especially relevant when discussing more complex function interactions, as it preserves certain desired mathematical properties like continuity.

Mathematical Proof

Mathematical proof is the rigorous method used to demonstrate the truth of a statement in mathematics. A proof involves a structured argument, using logical reasoning and established mathematical principles, that shows why a particular claim holds.

In proving uniform continuity for composed functions, such a proof begins with assumptions and definitions, such as what it means for each function to be uniformly continuous. We assume the properties hold true for each function separately and then build towards proving that these properties extend to their composition.

Here's how a typical proof unfolds:

• Start with the definitions and assumptions, such as given functions \(f\) and \(g\) are uniformly continuous.
• Identify known values (such as \(\epsilon\) and corresponding \(\delta\)) that define the uniform continuity for these functions.
• Use substitution and logical deductions to show that these properties result in uniform continuity for the function \(h(x) = g(f(x))\).
• Conclude with tying back the logical deductions to the final result, closing the proof with certainty of the claim.

This approach ensures the claim is irrefutably demonstrated, relying on the organized presentation of ideas and adherence to logical principles.

δ-ε Definition of Continuity

The \(\delta-\epsilon\) definition of continuity is fundamental in calculus and analysis, offering a precise way to talk about how functions behave. In simple terms, a function is continuous if you can make the output as close as you want to a particular value, by keeping the input sufficiently close to some fixed value.

In uniform continuity, the idea is extended slightly. Instead of focusing on a single point, we look at the domain as a whole. The simplest way to describe it:

• For every \(\epsilon > 0\), no matter how small, there exists a \(\delta > 0\) so that whenever any two points \(x\) and \(y\) in the domain satisfy \(|x - y| < \delta\), it follows that \(|f(x) - f(y)| < \epsilon\).
• Importantly, the \(\delta\) chosen works uniformly across the entire domain, not just around a specific point.

This uniformity is crucial when dealing with composed functions because it allows one to apply this consistent control over one function and then extend it over the other, ensuring the entire composition retains this type of continuity.