Explanation:

To determine whether the function f(x1, x2, ..., xn) = Max{|xi|} is uniformly continuous, continuous, or neither, we need to consider its properties.

Continuity:

First, let's determine if the function is continuous.

A function is continuous if it satisfies the epsilon-delta definition of continuity. According to this definition, for any given epsilon > 0, there exists a delta > 0 such that if the distance between two points in the domain is less than delta, then the difference between their function values is less than epsilon.

Let's consider two points in the domain of f, (x1, x2, ..., xn) and (y1, y2, ..., yn), such that the distance between these points is less than delta. The distance between two points in the Euclidean space can be calculated using the Euclidean distance formula:

d = sqrt((x1 - y1)^2 + (x2 - y2)^2 + ... + (xn - yn)^2)

Now, let's consider the difference between the function values at these points:

|f(x1, x2, ..., xn) - f(y1, y2, ..., yn)| = |Max{|xi|} - Max{|yi|}|

Since the maximum value of the absolute value of a set of numbers is the same as the maximum value of the numbers themselves, we can rewrite this as:

|f(x1, x2, ..., xn) - f(y1, y2, ..., yn)| = |Max{xi} - Max{yi}|

Since the maximum value of a set of numbers is a continuous function, the difference between the maxima can be made arbitrarily small by making the distance between the points small. Therefore, the function f(x1, x2, ..., xn) = Max{|xi|} is continuous.

Uniform Continuity:

Now, let's determine if the function is uniformly continuous.

A function is uniformly continuous if for any given epsilon > 0, there exists a delta > 0 such that if the distance between two points in the domain is less than delta, then the difference between their function values is less than epsilon, regardless of the specific points chosen.

To determine if the function is uniformly continuous, we need to show that for any epsilon > 0, we can find a delta > 0 such that for any two points in the domain with a distance less than delta, the difference between their function values is less than epsilon.

For the function f(x1, x2, ..., xn) = Max{|xi|}, consider two points (x1, x2, ..., xn) and (y1, y2, ..., yn) in the domain such that the distance between them is less than delta.

The maximum value of the absolute value of a set of numbers can change abruptly if the numbers themselves change abruptly. Therefore, as the distance between the points decreases, the difference between their function values can increase without bound.

For example, consider two points (1, 1, ..., 1) and (1+delta, 1, ..., 1) in the domain, where delta is a positive number. The distance between these points is delta, but the difference between their function values is also delta.

This shows that for any given epsilon > 0, it is not possible to find a