Question

Show that every polynomial of degree one on \(E^{n}\left({ }^{*}\right.\) or \(\left.C^{n}\right)\) is uniformly continuous.

Short answer

Every degree-one polynomial is uniformly continuous because it can be bounded by a linear expression.

Step-by-step solution

Define Uniform Continuity

A function \(f\) is uniformly continuous on a set \(S\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(x, y \in S\), if \(\|x - y\| < \delta\), then \(|f(x) - f(y)| < \epsilon\).

General Form of a First-Degree Polynomial

A polynomial of degree one in n variables, for real or complex numbers \(x_1, x_2, ..., x_n\), has the form \(f(x) = a_1x_1 + a_2x_2 + ... + a_nx_n + b\), where \(a_1, a_2, ..., a_n\) and \(b\) are constants.

Calculate the Difference for Uniform Continuity

For any two points \(x, y\) in \(E^n\) or \(C^n\), calculate the difference: \(|f(x) - f(y)| = |a_1(x_1 - y_1) + a_2(x_2 - y_2) + ... + a_n(x_n - y_n)|\). This can be bounded by \(|a_1||x_1 - y_1| + |a_2||x_2 - y_2| + ... + |a_n||x_n - y_n|\).

Apply the Euclidean Norm

Use the Euclidean norm to bound the expression: This equates to \(|f(x) - f(y)| \leq (|a_1| + |a_2| + ... + |a_n|)\|x - y\|\), where \|x - y\| is the Euclidean metric.

Find \(\delta\) for Uniform Continuity

For a given \(\epsilon > 0\), choose \(\delta = \frac{\epsilon}{|a_1| + |a_2| + ... + |a_n|}\). Then if \(\|x - y\| < \delta\), it follows directly that \(|f(x) - f(y)| < \epsilon\).

Conclusion

Since we can always find such a \(\delta\) for any \(\epsilon > 0\) and for any points \(x, y\) in \(E^n\) or \(C^n\), the function is uniformly continuous.

Key concepts

First-Degree Polynomial

A first-degree polynomial is one of the simplest types of polynomials. It's made up of a linear combination of variables with constant coefficients and an additional constant term. For instance, in an n-dimensional space, a first-degree polynomial can be written as:

• \( f(x) = a_1x_1 + a_2x_2 + ... + a_nx_n + b \)

Here:

• \( a_1, a_2, ..., a_n \) are coefficients, which are constant values for each variable.
• \( x_1, x_2, ..., x_n \) represent variables, which can be real or complex numbers.
• \( b \) is a constant term that translates the whole function along the polynomial's value axis.

First-degree polynomials are also known as linear functions because they graph as straight lines in two dimensions (a plane) or flat hyperplanes in higher dimensions. They play a crucial role in many mathematical fields and applications due to their simplicity and straightforward properties.

Euclidean Norm

The Euclidean norm, often referred to as the Euclidean length or Euclidean distance, is a way of measuring the "length" of a vector in Euclidean space. It's the most common norm used in mathematics due to its intuitive connection with distance and magnitude. For a vector \( x = (x_1, x_2, ..., x_n) \) in n-dimensional space, the Euclidean norm is given by:

• \( \|x\| = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} \)

This formula calculates the straight-line distance from the origin to the point \( (x_1, x_2, ..., x_n) \), which makes it a fundamental component in defining and understanding distances in space. In the context of polynomials and continuity, the Euclidean norm helps bound differences in function values, which is vital for proving uniform continuity. By using the inequality \( |f(x) - f(y)| \leq (|a_1| + |a_2| + ... + |a_n|)\|x - y\| \), we see how the norm serves as a critical tool for measuring the "closeness" of points in functional analysis.

Epsilon-Delta Definition

The epsilon-delta definition is a formal mathematical framework used to define the concept of continuity, especially uniform continuity. It encapsulates how changes in input (\( x \)) affect the output (\( f(x) \)) in controlled ways. According to this definition, a function \( f \) is uniformly continuous if:

• For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that, for all points \( x \) and \( y \) in the function's domain, if \( \|x - y\| < \delta \), then \( |f(x) - f(y)| < \epsilon \).

The key here is uniformity: \( \delta \) does not depend on the specific points \( x \) and \( y \). By choosing a suitable \( \delta \), as shown in the step-by-step solution, one can ensure that any variation in output is less than any predecided \( \epsilon \), confirming uniform continuity. The epsilon-delta definition is a cornerstone in calculus and analysis for understanding how functions behave across their entire domain, making it a powerful tool in both pure and applied mathematics.