The Continutiy and the Convergence.

[Tenshou]

Once upon a time there was a boy, neigh a man! He had trouble understand the connection between continuity and the different test for convergence. Sadly, he seen that they were connected and started to study, yet to no avail. Can someone please lend a helping hand on this quest for adventure, and excitement! (︶ω︶)

 

[mathman]

Try being more specific.

 

[Tenshou]

Well okay, Do you know how the uniform(not really just uniform, but most) convergence is (or can be) defined as $\lim_{k\to\infty} f_{k}\left(x\right) = f(x)$ *or* $f_{k}(x)\to f(x)$ uniformly as $k\to\infty$would mean that $\limsup_k\{|f_{k}(x)-f(x)| : \forall{x} \epsilon A\}$ which comes from a norm(or metric) and then there is the concept of continuity from calculus which states that $\lim_{x\to\infty} g(x) = L$ along with the other things that go with it (like the other concepts that would describe closeness, like the metric/norm it self and the epsilon/delta neighborhoods) why are these things so similar, are they the same?

 

[mathman]

There are many different types of limits. Uniform convergence is one, pointwise (not necessarily uniform) is another. There is also the concept of L_p convergence, p ≥ 1. epsilon/delta is a means of expressing convergence, not something different.

 

[Tenshou]

by $L_p$ you mean the those norm spaces, right? I mean, is the continuity convergence it self, or are they two completely different ideas. Like why do they look so similar?

 

[mathman]

L_p refers to normed spaces. I don't know what you mean by "continuity convergence".

 

[Tenshou]

No, I mean Is continuity, convergence? like the definition for the two look awfully the same, I still don't know the answer, but my assumption thus far has been that they(continuity and convergence) are the same as far as limits and families of functions go.

 

[jbunniii]

Continuity can be defined in terms of convergence: a function $f$ is continuous at a point $x$ if any sequence $x_n$ which converges to $x$ results in $f(x_n)$ converging to $f(x)$.

However, convergence is a more general concept. For example, we can talk about a series $\sum_{n=1}^{N} a_n$ converging to a value $A$ as $N \rightarrow \infty$, and this has nothing to do with continuity.

Or, we can define a function such as
$$f(x) = \begin{cases}
x^2 & \textrm{ if }x \neq 0 \\
1 & \textrm{ if }x = 0\\
\end{cases}$$
and we see that the limit as $x \rightarrow 0$ of $f(x)$ is $0$, even though the function is not continuous at $x = 0$.

 

[Tenshou]

Or, we can define a function such as
$$f(x) = \begin{cases}
x^2 & \textrm{ if }x \neq 0 \\
1 & \textrm{ if }x = 0\\
\end{cases}$$
and we see that the limit as $x \rightarrow 0$ of $f(x)$ is $0$, even though the function is not continuous at $x = 0$.

what do you mean it is not continuous at the point when x is zero, doesn't that mean it convergences point-wise, anyway? Or are you just stating a case when a function is discontinuous and yet still convergent?

 

[micromass]

what do you mean it is not continuous at the point when x is zero, doesn't that mean it convergences point-wise, anyway? Or are you just stating a case when a function is discontinuous and yet still convergent?

There is no "pointswise convergence" here. Jbunniii has just defined a function and stated that it is discontinuous. Furthermore, he said that this implied that the function is not sequentially continuous. That means, there is a sequence of real numbers that converges to a point, but whose images do not.

Pointswise convergence has nothing to do with this. Pointswise convergence deals with a sequence of functions and not a sequence with real numbers.

 

[Tenshou]

Pointswise convergence has nothing to do with this. Pointswise convergence deals with a sequence of functions and not a sequence with real numbers.

So, you saying that for sequences of real numbers(or numbers in general(?)) there cannot exist a convergent series, but this notion of convergent sequences only deals with functions?

 

[jbunniii]

So, you saying that for sequences of real numbers(or numbers in general(?)) there cannot exist a convergent series, but this notion of convergent sequences only deals with functions?

No, convergent sequences need not have anything to do with functions. For example, the sequence $a_n = 1/n$ is a sequence which converges to 0, but there is no function involved.

 

[Tenshou]

What about functions which have real numbers as convergent points. Do sequences of functions converge to real numbers? Vice versa?

 

[jbunniii]

What about functions which have real numbers as convergent points. Do sequences of functions converge to real numbers? Vice versa?

No, sequences of functions can converge to other functions, or they may converge at some points but not at others, or they may fail to converge at all. But I am not aware of any notion of convergence in which a sequence of functions may be said to converge to a number.

What we can do, however, is to define a metric on a set of functions, which is a way of measuring the distance between two functions. For example, on the space of integrable functions defined on $\mathbb{R}$, we may define
$$d(f,g) = \int_{-\infty}^{\infty} |f(x) - g(x)| dx$$
Then we may say that a sequence of functions $f_n$ converges to a function $f$ with respect to this metric. We define this to mean
$$\lim_{n \rightarrow \infty} d(f_n, f) = 0$$
Note that $d(f_n,f)$ is a sequence of numbers. The sequence of functions converges with respect to the metric if and only if the sequence of numbers $d(f_n,f)$ converges to zero. This is not the same as pointwise convergence! It is possible for a sequence $f_n$ to converge to $f$ with respect to the metric if it does not converge pointwise, and vice versa.