Question

Let \(\bar{p}_{0}, \bar{p}_{1}, \ldots, \bar{p}_{m}\) be fixed points in \(E^{n}\left({ }^{*}\right.\) or in another normed space \()\). Let $$f(t)=\bar{p}_{k}+(t-k)\left(\bar{p}_{k+1}-\bar{p}_{k}\right)$$ whenever \(k \leq t \leq k+1, t \in E^{1}, k=0,1, \ldots, m-1\) Show that this defines a uniformly continuous mapping \(f\) of the interval \([0, m] \subseteq E^{1}\) onto the "polygon" $$\bigcup_{k=0}^{m-1} L\left[p_{k}, p_{k+1}\right]$$ In what case is \(f\) one to one? Is \(f^{-1}\) uniformly continuous on each \(L\left[p_{k}, p_{k+1}\right] ?\) On the entire polygon?

Short answer

f is uniformly continuous, one-to-one if all points are distinct. f^{-1} is uniformly continuous on each segment, not on the entire polygon.

Step-by-step solution

Define the Conditions of f(t)

Let's understand how the function \( f(t) \) works. The function is defined piecewise for each segment \( k \leq t \leq k+1 \), where \( k = 0, 1, \ldots, m-1 \). For each interval, \( f(t) \) is a linear interpolation between the points \( \bar{p}_k \) and \( \bar{p}_{k+1} \). This means that \( f(t) = \bar{p}_k + (t-k)(\bar{p}_{k+1} - \bar{p}_k) \). This interpolation ensures it forms a continuous mapping on the interval \([0, m]\).

Show Uniform Continuity of f(t)

To show that \( f(t) \) is uniformly continuous, note that the function is linear on each subinterval \([k, k+1]\). Since a linear function in a normed space is Lipschitz continuous, there exists a constant \( L \) such that for any \( t_1, t_2 \in [k, k+1] \), \( \| f(t_2) - f(t_1) \| \leq L |t_2 - t_1| \). Consequently, this Lipschitz condition, consistent across each subinterval, proves that \( f(t) \) is uniformly continuous on \([0, m]\).

Determine One-to-One Mapping

The function \( f(t) \) will be one-to-one if, for every distinct \( t_1, t_2 \in [0, m] \), \( f(t_1) eq f(t_2) \). This condition is satisfied if the points \( \bar{p}_0, \bar{p}_1, \ldots, \bar{p}_m \) are all distinct, ensuring no overlapping in polygon segments.

Assess Uniform Continuity of f^{-1} on Segments

Within each segment \( L[\bar{p}_k, \bar{p}_{k+1}] \), the inverse \( f^{-1} \) maps a point on the line back to its corresponding \( t \). Since the mapping within each segment is linear by the definition of \( f(t) \), \( f^{-1} \) on each segment is uniformly continuous.

Consider Uniform Continuity of f^{-1} on Entire Polygon

The inverse \( f^{-1} \) is not uniformly continuous over the entire polygon unless there's a consistent way to measure 'distance' on the concatenated segments. Discontinuity might occur between segments where transitions aren't uniform, often making \( f^{-1} \) not uniformly continuous across the entire polygon.

Key concepts

Lipschitz Condition

The Lipschitz condition is a powerful property of a function that ensures it does not change too rapidly. When we say a function satisfies the Lipschitz condition, there is a constant \( L \) such that for all pairs of points \( x \) and \( y \) in a space, the inequality \( \| f(x) - f(y) \| \leq L \| x - y \| \) holds. This implies that the function is controlled in terms of how fast it can change.
For the function \( f(t) \), which is linear on each interval \([k, k+1]\), the Lipschitz condition is satisfied due to its linear nature in any normed space. The uniform continuity we obtain from this is because the Lipschitz constant \( L \) holds consistently across all intervals.
This maintains a stable bound on the function's growth and makes \( f(t) \) uniformly continuous over the entire domain \([0, m]\).
This is a critical aspect as it assures that small changes in \( t \) lead to small changes in \( f(t) \), making it predictable and manageable.

Linear Interpolation

Linear interpolation is a method of estimating new data points within the range of a discrete set of known data points. In essence, it helps us "draw straight lines" between given points to usually provide a smooth transition from one to the next.
In the context of the exercise, the function \( f(t) \) is defined such that it linearly interpolates between fixed points \( \bar{p}_k \) and \( \bar{p}_{k+1} \) for \( k \leq t \leq k+1 \).

• This means that between any two adjacent fixed points, \( f(t) \) represents the straight line connecting them.
• The formula \( f(t) = \bar{p}_k + (t-k)(\bar{p}_{k+1} - \bar{p}_k) \) precisely guarantees that this segment is straight.
• This ensures a continuous and piecewise linear mapping from \([0, m]\) to the polygonal path.

The use of linear interpolation here helps in constructing a simple yet effective bridge between points that maintains continuity over the domain. This characteristic is crucial for defining \( f(t) \) as a coherent mapping on the interval.

Inverse Function

The concept of an inverse function revolves around reversing the effect of a function. If a function \( f \) maps an element \( x \) to \( y \), then its inverse \( f^{-1} \) should map \( y \) back to \( x \).
In this exercise, \( f(t) \) describes a mapping of \([0, m]\) onto a polygon. A crucial part of this is determining the conditions under which \( f \) is one-to-one (injective) and when its inverse \( f^{-1} \) is uniformly continuous.

• \( f(t) \) is one-to-one if each \( \bar{p}_k \) is distinct, meaning no overlap in polygon segments.
• Each individual section or segment between \( \bar{p}_k \) and \( \bar{p}_{k+1} \) has a correspondence between \( t \) and \( f(t) \), allowing for \( f^{-1} \) to be defined there.
• Because the mapping within each segment is linear, \( f^{-1} \) on each individual segment is uniformly continuous. However, across the entire polygon, unless the joins between segments are also controlled, \( f^{-1} \) may lose this property.

Understanding inverse functions is fundamental to decoding the overall behavior of mappings and ensuring that they can be reversed smoothly in certain conditions.

Normed Space

A normed space is essentially a vector space on which a function (called a norm) assigns a strictly positive length or "size" to each vector in the space, other than the zero vector. This is a mathematical environment where distances are measured consistently.
The norm \( \| \cdot \| \) provides the framework within which continuity and convergence are analyzed. A function defined on a normed space can utilize this metric to discuss various properties like uniform continuity and Lipschitz continuity.
In our discussion:

• The function \( f(t) \) resides in a normed space, allowing for the meaningful application of the Lipschitz condition due to the presence of a norm.
• This space aids in defining linear interpolations and uniform continuity through clearly understood measures of distance.
• Understanding \( f(t) \)'s behavior in such a space helps ascertain properties such as how 'far apart' points are and how the function changes across segments.

This clear methodology for assessing distance and continuity in a normed space is what underpins much of the theoretical discussion of continuity and interpolation in higher mathematics.