Question

Prove that the limit \(\mathbf{f}(t)\) of a uniformly convergent sequence of functions \(\left\\{f_{n}(t)\right\\}\) continuous on \([a, b]\) is itself a function continuous on \([a, b] .\) Hint. Clearly $$ \left|f(t)-f\left(t_{0}\right)\right| \leqslant\left|f(t)-f_{n}(t)\right|+\left|f_{n}(t)-f_{n}\left(t_{0}\right)\right|+\left|f_{n}\left(t_{0}\right)-f\left(t_{0}\right)\right| $$ where \(t, t_{0} \in[\mathrm{a}, \mathrm{b}]\). Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large \(\mathrm{n}\). Then use the continuity of \(f_{n}(t)\) to make the second term small for \(t\) sufficiently close to \(t_{0}\).

Short answer

The limit function \( f(t) \) is continuous on \([a, b]\).

Step-by-step solution

Understanding Uniform Convergence

For a sequence of functions \( \{ f_n(t) \} \) to converge uniformly to \( f(t) \) on \([a, b]\), for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n > N \) and all \( t \in [a, b] \), \( |f_n(t) - f(t)| < \epsilon/3 \). This is a fundamental property of uniform convergence that helps ensure \( f(t) \) inherits some properties from \( f_n(t) \).

Examining the Continuity of \( f(t) \)

To prove \( f(t) \) is continuous, we analyze \( |f(t) - f(t_0)| \) for any \( t, t_0 \in [a, b] \). The hint suggests this inequality:\[ |f(t) - f(t_0)| \leq |f(t) - f_n(t)| + |f_n(t) - f_n(t_0)| + |f_n(t_0) - f(t_0)| \] Each term can be made small, therefore making \( |f(t) - f(t_0)| \) small, proving continuity at \( t_0 \).

Applying Uniform Convergence to the First and Third Terms

Since the sequence \( \{ f_n(t) \} \) converges uniformly to \( f(t) \), for \( n > N \), we can make \( |f(t) - f_n(t)| < \epsilon/3 \) and \( |f_n(t_0) - f(t_0)| < \epsilon/3 \) for any \( \epsilon > 0 \) and all \( t, t_0 \in [a, b] \). These terms are small by the definition of uniform convergence, irrespective of \( t \).

Utilizing the Continuity of \( f_n(t) \) for the Second Term

Given that each \( f_n(t) \) is continuous on \([a, b]\), for the second term \( |f_n(t) - f_n(t_0)| \), for each \( n > N \), we choose \( \delta > 0 \) such that if \( |t - t_0| < \delta \), then \( |f_n(t) - f_n(t_0)| < \epsilon/3 \). This leverages the continuity of \( f_n(t) \) specific to each chosen \( n \).

Combining the Terms and the Limit Process

Considering the inequality:\[ |f(t) - f(t_0)| \leq |f(t) - f_n(t)| + |f_n(t) - f_n(t_0)| + |f_n(t_0) - f(t_0)| \]With \( n > N \), we've established that each term is less than \( \epsilon/3 \), making the combined sum less than \( \epsilon \):\[ |f(t) - f(t_0)| < \epsilon \]This holds for any \( \epsilon > 0 \) and illustrates that as \( t \rightarrow t_0 \), \( f(t) \rightarrow f(t_0) \), thus \( f(t) \) is continuous on \([a, b]\).

Key concepts

Continuous Functions

In mathematics, a function is continuous at a point if a small change in input results in a small change in output. This idea means there are no sudden jumps or gaps in the function's graph. For a function to be continuous on an interval, it needs to be continuous at every point within that interval.

A function \( f(t) \) is continuous at a point \( t_0 \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |t-t_0| < \delta \), it follows that \( |f(t) - f(t_0)| < \epsilon \).

This definition ensures that no matter how small a change we wish to see in \( f(t) \), there's a small enough region around \( t_0 \) where all values of \( f(t) \) are close enough to \( f(t_0) \). In the case of uniform convergence, we know that the limit function can inherit this property of continuity from the approximating sequence of functions.

Sequence of Functions

A sequence of functions is simply a list of functions indexed by natural numbers. Each member of the sequence is defined on the same interval, such as \([a, b]\). For example, a sequence might look like \( f_1(t), f_2(t), f_3(t), \ldots \).

Understanding how such sequences behave as they "approach" a particular function is crucial in real analysis. With uniform convergence, the entire sequence comes steadily closer to a target function \( f(t) \) over the entire interval \([a, b]\), not just at individual points. This means that the entire graph of the functions tends to match that of the limit function closely as \( n \rightarrow \infty \).

Uniform convergence is particularly important because it ensures that the properties like continuity of the functions in the sequence are transferred to the limit function.

Limit of a Function

In real analysis, the limit of a function refers to the value that the function "approaches" as its input nears a particular point. When considering a sequence of functions \( \{f_n(t)\} \) that converges uniformly to \( f(t) \), the limit of the sequence at each point is the value to which \( f_n(t) \) approaches as \( n \rightarrow \infty \).

For uniformly convergent sequences, the critical point is that the limit function \( f(t) \) is approached in the same way across the entire interval \([a, b]\), not just at isolated points. This uniformity of convergence ensures that attributes like continuity can pass onto the limit function \( f(t) \) even if each \( f_n(t) \) is defined separately. Thus, the idea of limits in this context is pivotal for ensuring the integrity of function behaviors as we transition from sequences to single functions.

Real Analysis

Real analysis is a branch of mathematics dealing with real numbers and real-valued functions. It mainly involves the rigorous study of sequences, limits, continuity, and other properties derived from them. One of the key aspects in real analysis is understanding the behavior of functions and sequences within a given interval.

In this exercise, real analysis provides us with the tools to rigorously demonstrate that certain properties (like continuity) can be preserved when moving from sequences (which we can think of as an infinite list of functions) to a single limit function. This exercise uses a combination of theories and definitions from real analysis to show how uniform convergence affects the property of continuity for a sequence of continuous functions. By understanding these concepts, real analysis helps ensure that the behaviors of sequences like \( \{f_n(t)\} \) are predictable and reliable when taking limits.