Uniform Convergence, Cauchy Sequence, Uniform Continuous

Question

I kept on coming across the topic of Uniform Convergence. Wonder if someone could help me with the following questions:

1) What is the object of reference of Uniform Convergence? Series? Sequence? Functions? All of the above and others?

2) How does Uniform Convergence differ or relate to Cauchy Sequence? Seems like the same.

3) How does Uniform Convergence differ or relate to Uniform Continuous?

4) Lastly, how does Uniform Convergence relate to Right Continuous?

Thank you!

Answer 1

1) What is the object of reference of Uniform Convergence? Series? Sequence? Functions? All of the above and others?

The object is a sequence/series of functions.

2) How does Uniform Convergence differ or relate to Cauchy Sequence? Seems like the same.

These are totally different concepts. A Cauchy sequence is a property of a sequence of numbers, not functions:

A Cauchy sequence is a s sequence that the numbers in it are getting closer and closer to each-other (for every distance $\varepsilon$ , there exists some place at the sequence, $N$, that after $a_N$ all of the elements are close to each-other more than a distance of $\varepsilon$).

Uniform convergence is something different: imagine the sequence $f_n(x) = x^n $ in the interval $I=[0,1)$. For every $x\in I$, $x^n \to 0$ so the sequence of function tends towards $f(x)=0$. Meaning that for every point $x\in I$ and for every distance $\varepsilon$ there is a place in the sequence $N$ such that $\forall n>N : |f_n(x)-0|=x^n < \varepsilon $. That is the definition of convergence of a sequence of functions - no surprise here. But when you think about it, for every N you give, there are some points in the interval that $x^N>\varepsilon$, you just need take points sufficiently close to $1$. Meaning there is no place in the sequence that after that place the hole function is close to the limit, each point has its own place in the sequence. Uniform convergence is when for every distance you DO have a place in the sequence that the whole function is closer to the limit function than $\varepsilon$.

This is a property of a sequence of functions and an interval. For $f_n(x) = x^n $ (the same function) but with the interval $I=[0,0.5]$ there is uniform convergence.

3) How does Uniform Convergence differ or relate to Uniform Continuous?

Again, Uniform Convergence is a property of a sequence of functions and an interval, whereas Uniform Continuous is a property of a function and an interval (not a sequence of functions).

4) Lastly, how does Uniform Convergence relate to Right Continuous?

See answer to section (3)

Answer 2

To talk about uniform convergence we need a set $X$ and the set $F(X)$ of functions $X \to \mathbb R$

1. It might be the sequence of functions $f , f_1,f_2, \ldots$ tends uniformly to the function $f$. This means the sequence of real numbers $\sup |f(x)-f_n(x)|$ tends to zero. This makes sense whether or not $X$ has a metric.

2. You need a metric to talk about a Cauchy sequence. If $X$ is a metric space it might be the sequence $x_1,x_2,\ldots$ is Cauchy. But even if $X$ has no metric $F(X)$ does! That metric is $d(f,g)=\sup |f(x)-g(x)|$. So it makes sense to ask whether or not a sequence $f_1,f_2,\ldots$ is Cauchy.

3. A continuous element of $g \in F(X)$ might be uniformly continuous. But most elements of $F(X)$ are not continuous in the first place! But that's fine because we can have $f_n \to f$ uniformly even if none of $f_n$ or $f$ are continuous.

4. There is no immediate link between right-continuity and uniform convergence. There is a link between right-continuity and uniform continuity however. It only makes sense to say $f \colon X \to \mathbb R$ is continuous if $X= \mathbb R$. But in that case uniform continuity implies continuity implies right continuity.