Question

The function \(f:\left(\mathbb{R}^{*}\right)^{2} \rightarrow[0,+\infty]\) defined by \(f(t, s)=|t-s|\) for \(t, s \in \mathbb{R}\), \(f(\infty, t)=f(t, \infty)=+\infty\) for \(t \in \mathbb{R}\), and \(f(\infty, \infty)=0\) is lower semicontinuous.

Short answer

The function is lower semicontinuous for all defined cases.

Step-by-step solution

Understand the Function

The function \( f \) is defined for pairs \((t, s)\) and maps to a non-negative value. It is essentially the absolute difference \(|t - s|\) when both \(t\) and \(s\) are real numbers. Special cases are defined at positive and negative infinity, where the value is \( +\infty \) except when both arguments are infinity, which returns \( 0 \).

Recall Lower Semicontinuity

A function is lower semicontinuous at a point if for every sequence \((t_n, s_n)\) converging to \((t, s)\), \( f(t, s) \leq \liminf_{n \to fty} f(t_n, s_n) \). This implies the function value is at most the limit inferior of the function values along a sequence converging to the point.

Check Lower Semicontinuity for Real Inputs

Consider a sequence \( (t_n, s_n) \to (t, s) \) with \( t, s \in \mathbb{R} \). We have \(f(t, s) = |t - s|\). For any sequence, \( |t_n - s_n| \to |t - s| \), thus \( \lim_{n \to fty} f(t_n, s_n) = f(t, s) \). Since the limit equals the actual value, this inherently satisfies the lower semicontinuous condition.

Check Special Cases for Infinity

For sequences converging to infinity, consider \( t_n \to fty \) or \( s_n \to fty \). The function \( f(t_n, fty) = +fty \) and similarly for the other case, which is consistent with lower semicontinuity since the value cannot exceed \(+fty\). When both arguments approach \(fty\), \( f(fty, fty) = 0\) fits because sequences where both go to infinity should yield a zero value based on the definition.

Conclusion

For all scenarios of input pairs \((t,s)\) including special cases involving infinity, the function \( f \) satisfies the criteria for lower semicontinuity. This includes real number inputs and infinite boundaries as well.

Key concepts

Real Analysis

Real analysis is a core part of mathematics that deals with real numbers and real-valued sequences and functions. It provides tools and concepts to understand the properties of real functions, such as continuity, limits, and differentiability. In this context, we are dealing with a function that is defined over real numbers and infinity, characteristics that belong to the realm of real analysis. The function in question, defined by the absolute difference between two values, is a classic example of exploring behaviors in real analysis due to its inclusion of real number arguments and concepts like limits and continuity. Real analysis helps us in:

• Understanding Limits: Limits allow determining how a function behaves as inputs approach certain points or regions, like at infinity.
• Exploring Continuity: We use concepts from real analysis to investigate when and where a function is continuous or semicontinuous, as in this function's case with lower semicontinuity.
• Handling Infinite Values: Real analysis provides a framework for handling infinite limits and boundaries.

In analyzing our function, these fundamental concepts from real analysis come into play, particularly when considering how sequences behave as they converge to points or infinity.

Infinite Boundaries

Infinite boundaries present a unique challenge in mathematical analysis. With functions that approach or include infinity, we must consider how they behave differently from ones confined to finite values. In this exercise, infinity is incorporated into the function's domain and range:

• For real numbers, the function behaves predictably by computing absolute differences.
• When either argument is infinity, the function returns "+∞," highlighting a boundary at infinity.
• When both arguments are infinity, the function interestingly returns zero, a special case explicitly defined in the problem.

These behaviors around infinite boundaries require special attention:

• Consistent Definitions: Special cases need precise definitions, as seen in the handling of infinite boundaries where certain properties, like returning zero for both arguments being infinity, must be logical and consistent.
• Handling Infinity in Calculations: Since mathematical operations involving infinity can have non-intuitive results, we ensure functions align with real-world understanding.
• Evaluating Limits: As sequences approach infinity, how functions behave is integral to real and complex analysis.

Through careful definition and evaluation, infinite boundaries are crucial for understanding the broader picture of function behaviors in real analysis.

Limit Inferior

The concept of the limit inferior is vital in determining lower semicontinuity, especially when we consider sequences approaching a point. The limit inferior of a sequence is the greatest lower bound of its subsequential limits, essentially the smallest value the sequence can infinitely approach. When determining lower semicontinuity:

• Importance of Limit Inferior: For lower semicontinuity, a function's value at a point should be less than or equal to the limit inferior of the function values along a converging sequence.
• Assessing Convergence: By comparing function values with the limit inferior, we ascertain that the function doesn't suddenly dip below the expected value.
• Calculating with Infinity: Include sequences involving "+∞" to verify that they align with the function's special definitions, as seen when approaching points at infinity.

In this exercise, limit inferior ensures the function’s behavior, both for real values and infinite values, aligns with the definition of lower semicontinuity. When the sequences converge either to finite points or infinity, the limit inferior helps verify the bounds and continuity conditions accurately.