Need explanation of theorems on Uniform continuity

In summary, we learned two theorems about Uniform Continuity, one involving a function on an arbitrary set and the other involving a function on a closed and bounded interval. These theorems state that given certain conditions, a function that is continuous will also be uniformly continuous. This is because the closed and bounded interval limits how much the function can spread out, and the Cauchy sequence guarantees that the terms of the sequence eventually get close together, allowing us to control the distance between points. These theorems are stated in this way to avoid counterexamples and ensure that the function remains uniformly continuous.
  • #1
VreemdeGozer
12
0
I'm taking my first course in Analysis, and we learned a couple of theorems about Uniform Continuity. I have been able to visualize most of what's been going on before, but I need some help with the following:

E [itex]\subseteq[/itex] ℝ, f: E [itex]\rightarrow[/itex] ℝ uniform continuous. if a sequence xn is Cauchy [itex]\Rightarrow[/itex] f(xn) is Cauchy

I is a closed, bounded interval, f: I [itex]\rightarrow[/itex] ℝ. if f is continuous on I [itex]\Rightarrow[/itex] f is uniformly continuous on I

We are using the international version of: An Introduction to Analysis by William R. Wade, fourth edition.

I'm really looking for a visual explanation, but if anyone can explain why it works in words, that's fine too.
 
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  • #2
f is continuous so |f(x+h)-f(x)|<epsilon(x)
epsilon can vary with x
since I is closed and bounded max epsilon(x) exist thus
|f(x+h)-f(x)|<max epsilon(x)
hence f is uniform continuous

So the closed and bounded I allows us to deduce uniform continuity from continuity by limiting how much f can spread out.
 
  • #3
I'll try for an intuitive idea of why these theorems are true. It's helpful that they're actually very, very similar ideas (if I am interpreting your statements correctly).

As is often true with theorems, it's helpful to understand why they're stated the why they are. Deleting hypotheses often helps us find counterexamples for the modified statement, and this will help us understand why the theorem is stated the way it is (and in turn make the statement more intuitive).

In the second example, we need a closed and bounded interval. This is good because that means any function continuous on that interval has a maximum and minimum. Hence we know that the distance between the function at any two points is bounded by, say, some number n. Given continuity, this is enough to get uniform continuity, since uniform continuity just requires that we can in some sense control the distance between the function at any two points by controlling the distance between the two points. It may be helpful to think about why we need a closed interval. To see why we want a closed interval, think about the topologist's sine curve. That would be problematic for uniform continuity. Conversely, what about if the interval was unbounded?

The Cauchy sequence bit is pretty similar. It helps to think about why we need it to be a Cauchy sequence. All that adds is that "after a while the rest of the terms are close together". But this is sort of like how we required that the interval in the above theorem be closed and bounded, so the logic is similar.

Of course these aren't proofs but I tried to give some of the intuition you seem to want. Hope it helps!
 

1. What is uniform continuity?

Uniform continuity is a mathematical concept that describes the behavior of a function by analyzing its rate of change over a given interval. It ensures that the function changes gradually and consistently, without any sudden jumps or breaks.

2. How is uniform continuity different from regular continuity?

Uniform continuity is a stronger form of continuity compared to regular continuity. While regular continuity only considers the behavior of a function at a single point, uniform continuity considers the behavior of the function over an entire interval. This means that a function can be uniformly continuous even if it is not continuous at a particular point.

3. What is the epsilon-delta definition of uniform continuity?

The epsilon-delta definition of uniform continuity states that for any given epsilon (ε) greater than 0, there exists a delta (δ) greater than 0 such that for all x and y in the domain of the function, if the distance between x and y is less than delta, then the distance between f(x) and f(y) is less than epsilon.

4. How do you prove that a function is uniformly continuous?

In order to prove that a function is uniformly continuous, you must show that it satisfies the epsilon-delta definition of uniform continuity. This can be done by choosing a specific epsilon and finding a corresponding delta that satisfies the definition for all x and y in the domain.

5. What are some examples of uniformly continuous functions?

Some examples of uniformly continuous functions include polynomials, trigonometric functions, and exponential functions. These functions have a smooth and consistent rate of change over their entire domain, making them uniformly continuous.

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