Pointwise and uniform convergence are two important ideas in understanding how sequences of functions behave, especially when it comes to their continuity.

Pointwise Convergence:

A sequence of functions, written as $\{f_n\}$, converges pointwise to a function $f$ on a set $D$ if, for every point $x$ in $D$, the limit

$$\lim_{n \to \infty} f_n(x) = f(x)$$

exists. This means that as we look at each point $x$ in the set, the sequence of functions gets closer and closer to the function $f$.

However, here’s the catch: Just because each $f_n$ is continuous (meaning it does not have any jumps or breaks at that point) doesn’t mean that the limit function $f$ will also be continuous.

To understand this better, let’s look at an example:

Imagine a sequence of functions defined like this:

$$f_n(x) = \begin{cases} 1 & \text{if } x \in [0, 1 - \frac{1}{n}] \\ 0 & \text{if } x \in (1 - \frac{1}{n}, 1] \end{cases}$$

For any value of $n$, the function $f_n$ is continuous on the interval from 0 to 1. But as $n becomes larger,$f_n(x)$ gets closer to this function:

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

In this case, $f$ is not continuous at $x = 1$. So even though the sequence of functions converges pointwise to $f$, this doesn’t ensure that $f$ itself is continuous.

Uniform Convergence:

Now, uniform convergence is a stronger idea. A sequence of functions $\{f_n\}$ converges uniformly to a function $f$ on a set $D$ if:

$$\lim_{n \to \infty} \sup_{x \in D} |f_n(x) - f(x)| = 0.$$

This means that the difference between $f_n(x)$ and $f(x)$ is getting smaller and smaller across the entire set $D$, not just at individual points.

In simpler terms, if the convergence is uniform, it guarantees that if each function $f_n$ is continuous, then the limit function $f$ will also be continuous.

For example, consider these functions:

$$f_n(x) = \frac{x}{n}$$

on the interval $[0, 1]$. Each of these functions is continuous, and they converge uniformly to:

$$f(x) = 0.$$

Since this convergence is uniform, we can confidently say that the limit function $f(x)$ is continuous too.

Key Differences:

1. Pointwise Convergence:

   • Definition: The sequence $f_n$ converges pointwise to $f$ on $D$ if $\lim_{n \to \infty} f_n(x) = f(x)$ for each $x$ in $D$.
   • Continuity: Pointwise convergence does not guarantee that the limit function is continuous.

2. Uniform Convergence:

   • Definition: The sequence $f_n$ converges uniformly to $f$ on $D$ if $\lim_{n \to \infty} \sup_{x \in D} |f_n(x) - f(x)| = 0$.
   • Continuity: Uniform convergence keeps the limit function continuous if each $f_n$ is continuous.

Understanding these two types of convergence is essential for anyone studying calculus, especially when looking at how functions behave over time.

To sum it up:

• Pointwise convergence looks at individual points and often leads to issues with continuity.

• Uniform convergence provides a more solid structure, ensuring continuity is maintained if all the individual functions are continuous.

In conclusion, when you are working with sequences of functions, it is very important to check the type of convergence. This can greatly affect the continuity and behavior of the limit function. Pointwise convergence is more relaxed, while uniform convergence guarantees a tighter control over continuity, leading to clearer and more predictable results.