Continuity And Pointwise Convergence Doesn't Imply Contiunity

Introduction

In the realm of real analysis, continuity and pointwise convergence are two fundamental concepts that are often studied in conjunction with each other. However, a common misconception is that if a sequence of continuous functions converges pointwise to a function, then the limit function is also continuous. In this article, we will explore this notion and provide a counterexample to demonstrate that pointwise convergence of continuous functions does not necessarily imply continuity of the limit function.

Pointwise Convergence

Before we dive into the main topic, let's briefly recall the definition of pointwise convergence. A sequence of functions $f_n$ is said to converge pointwise to a function $f$ on a set $E$ if for every $x \in E$ and every $\epsilon > 0$, there exists a natural number $N$ such that for all $n \geq N$, $|f_n(x) - f(x)| < \epsilon$. In other words, the sequence of functions converges pointwise to $f$ if the sequence of numbers $f_n(x)$ converges to $f(x)$ for each individual $x \in E$.

Counterexample

To show that pointwise convergence of continuous functions does not imply continuity of the limit function, we will provide a counterexample. Let's consider the following sequence of functions on the interval $[0,1]$:

$$f_n(x) = \begin{cases} 0 & \text{if } x \in [0,1] \text{ and } x \notin \left\{ \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n} \right\} \\ 1 & \text{if } x \in \left\{ \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n} \right\} \end{cases}$$

Each function $f_n$ is continuous on $[0,1]$ since it is a piecewise constant function with finitely many discontinuities. Moreover, the sequence $f_n$ converges pointwise to the function $f$ defined by:

$$f(x) = \begin{cases} 0 & \text{if } x \in [0,1] \text{ and } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$

To see this, let $x \in [0,1]$ and $\epsilon > 0$. Choose $N$ such that $\frac{1}{N} < \epsilon$. Then for all $n \geq N$, we have $|f_n(x) - f(x)| < \epsilon$ since $f_n(x) = 0$ and $f(x) = 0$ for all $x \neq 0$.

However, the limit function $f$ is not continuous at $x = 0$. In fact, $f$ has a jump discontinuity at $x = 0$ since $\lim_{x \to 0^-} f(x) = 0$ and $\lim_{x \to 0^+} f(x) = 1$.

Conclusion

In conclusion, we have shown that pointwise convergence of continuous functions does not necessarily imply continuity of the limit function. The counterexample provided demonstrates that even if a sequence of continuous functions converges pointwise to a function, the limit function may still have discontinuities. This result highlights the importance of carefully analyzing the properties of the limit function in real analysis.

Implications

The result we have obtained has several implications in real analysis. Firstly, it shows that pointwise convergence is a weaker notion than uniform convergence. In fact, if a sequence of continuous functions converges uniformly to a function, then the limit function is also continuous. Secondly, it highlights the importance of considering the properties of the limit function in real analysis, rather than just focusing on the convergence of the sequence of functions.

Further Reading

For those interested in exploring this topic further, we recommend the following references:

• Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• Bartle, R. G. (1964). The Elements of Real Analysis. John Wiley & Sons.
• Royden, H. L. (1988). Real Analysis. Prentice Hall.

Q: What is the difference between pointwise convergence and uniform convergence?

A: Pointwise convergence is a weaker notion of convergence that requires the sequence of functions to converge to the limit function at each individual point. Uniform convergence, on the other hand, requires the sequence of functions to converge to the limit function at all points simultaneously. In other words, pointwise convergence allows for the possibility of different rates of convergence at different points, while uniform convergence requires a uniform rate of convergence across all points.

Q: Can you provide another example of a sequence of continuous functions that converges pointwise to a discontinuous function?

A: Yes, consider the sequence of functions $f_n(x) = \sin(n \pi x)$ on the interval $[0,1]$. Each function $f_n$ is continuous on $[0,1]$ since it is a trigonometric function. Moreover, the sequence $f_n$ converges pointwise to the function $f(x) = 0$ on $[0,1]$. However, the limit function $f$ is not continuous at $x = \frac{1}{2}$ since $\lim_{x \to \frac{1}{2}^-} f(x) = 1$ and $\lim_{x \to \frac{1}{2}^+} f(x) = -1$.

Q: What are some common misconceptions about pointwise convergence?

A: One common misconception is that pointwise convergence implies uniform convergence. As we have seen, this is not the case. Another misconception is that pointwise convergence requires the sequence of functions to converge to the limit function at all points simultaneously. In fact, pointwise convergence only requires convergence at each individual point.

Q: Can you provide a counterexample to the statement that pointwise convergence of continuous functions implies continuity of the limit function?

A: Yes, consider the sequence of functions $f_n(x) = \begin{cases} 0 & \text{if } x \in [0,1] \text{ and } x \notin \left\{ \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n} \right\} \\ 1 & \text{if } x \in \left\{ \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n} \right\} \end{cases}$ on the interval $[0,1]$. Each function $f_n$ is continuous on $[0,1]$ since it is a piecewise constant function with finitely many discontinuities. Moreover, the sequence $f_n$ converges pointwise to the function $f(x) = \begin{cases} 0 & \text{if } x \in [0,1] \text{ and } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$. However, the limit function $f$ is not continuous at $x = 0$ since $\lim_{x \to 0^-} f(x) = 0$ and $\lim_{x \to 0^+} f(x) = $.

Q: What are some real-world applications of pointwise convergence?

A: Pointwise convergence has numerous real-world applications in fields such as signal processing, image analysis, and machine learning. For example, in signal processing, pointwise convergence is used to analyze the behavior of signals over time. In image analysis, pointwise convergence is used to study the behavior of images over space. In machine learning, pointwise convergence is used to analyze the behavior of models over time.

Q: Can you provide a summary of the key points discussed in this article?

A: Yes, the key points discussed in this article are:

• Pointwise convergence is a weaker notion of convergence that requires the sequence of functions to converge to the limit function at each individual point.
• Uniform convergence is a stronger notion of convergence that requires the sequence of functions to converge to the limit function at all points simultaneously.
• Pointwise convergence does not imply uniform convergence.
• Pointwise convergence of continuous functions does not necessarily imply continuity of the limit function.
• Pointwise convergence has numerous real-world applications in fields such as signal processing, image analysis, and machine learning.