Pointwise Limits, Uniform Limits, and Continuity

Before diving into the muck head first, take a moment to go over some of the basic ideas involving continuity -particularly, uniform continuity. We will lightly cover a few concepts of continuity to provide a base for our discussion.

We know from basic calculus and our study of integration, differentiation, and continuity that:

{f}=f_{1} +f_{2}+f_{3}+f_{4}+f_{5}+\cdots +f_n

on an interval {I}, then (taken from Elementary Real Analysis by Andrew Bruckner).

(1) If f_{1},f_{2},f_{3},\cdots,f_n are continuous on {I}, so is {f}

(2) If f_{1},f_{2},f_{3},\cdots,f_n are differentiable on {I}, then so is {f}. And it follows that,

f^{'}=f_{1}^{'}+f_{2}^{'}+\cdots+f_{n}^{'}

(3) If f_{1},f_{2},f_{3},\cdots,f_{n} are integrable on {i}, then so is {f} And we have,

\int_a^b\! f(x){dx}=\int_a^b\! f_{1}(x){dx}+\cdots \int_a^b\!f_{n}(x){dx}

Everything seems to be as expected, right? So, what about adding an infinite number of  functions f_{n}. That is, what would happen, or what can we say about if we had to consider an infinite series of functions?

{f}=\sum_{k=0}^{\infty}{f_{n}}

For points (1)-(3), imagine, for a moment, the nature of an infinite sum. Will it produce the same results?

To spoil the ending, the truth is: (1)-(3) do not necessarily hold as we “stretch” the terms out to infinity. We will is that there is a different type of condition that must absolutely hold for our reproducing the same results as we did in (1)-(3) for an infinite iteration of functions.

POINTWISE LIMITS

As is obvious, we are dealing with sequences of real-valued functions. The nature of how these functions work as a series is, however, not so obvious. In fact, we know what it means for a series of numerical values to converge, but doe the same thing work for functions? This question is among many that we must answer. In fact, is the limit of a sequence of functions still a function? What about differentiation? If each function {f_{n}} is differentiable, is $latex \lim_{n \to +\infty} f_{n}$ differentiable? The same goes for integration. And what about continuity? If each of our functions f_{n} is continuous on some \mathbb{D}, is the limit of these functions as well?

We must cash-out some definitions, look at some properties, and, then, see what theorems follow. From this, we can work some problems and develop an understanding before moving to uniform convergence.

Now, I will be drawing on my own notes for this post, but I recommend that you look at this site as a means to gain some further clarification. You can also find some references and discussions here.

Definition 1:

Let {f_n} be a sequence of of functions defined on some domain \mathbb{D}. If the \lim\limits_{n \to \infty} f_{n}(x) \in \mathbb{R} for all x in \mathbb{D} we say that the sequence converges pointwise on our domain. The limit defined a functions {f} on \mathbb{D} by the equation,

f(x)= \lim\limits_{n \to \infty} f_{n}(x)

and we say that f_{n} \rightarrow {f}.

What stands out about this definition? Well, to see it better, lets write it out in a more logical fashion:

We assert that f_{n} pointwise to {f} on \mathbb{D} if \forall{x} \forall {\epsilon}>0 \exists{N}\in\mathbb{N} :\forall {n} \geq{N} \rightarrow |f_{n}-{f}|<\epsilon notice that N appears after the quantification of x and \epsilon. Thus, N may depend on the values of these terms. Make special note as to the quantification of x here as well.

This sets us up for the next big question: what do we make of,

\sum\limits_{k=1}^{\infty}f_{k}(x)

Definition 2:

For each x in \mathbb{D} and n$latex\in\mathbb{N}$ we let,

S_{n}(x)=f_{1}(x)+\cdots+f_{n}(x).

If the limit of partial sums is taken (i.e. \lim\limits_{n \to \infty}S_{n}(x)) and if such is a real number, we say that the series converges at x for the limit above. If the series converges for all x\mathbb{D}, we say that the series converges pointless on D to the function {f} defined by,

f(x)=\sum\limits_{k=1}^{\infty} f_{k}(x)= \lim\limits_{n \to \infty}\sum\limits_{k=1}^{n}

In general we cannot use piecewise limits to provide positive answers to the questions of {f}  being continuous if f_{n} is on some interval, nor can we do it for differentiation or integration.

For example, consider the following function f_{n}=x^n for x in [0,1]. Each of the f_{n} are continuous on [0,1]. But what about at each x? Take consideration of x\in(0,1) the limit \lim\limits_{n \to \infty}f_{n}=0 and, yet, we get \lim\limits_{n \to \infty}f_{n}(1)=1 ! Look at the graph of the function to “see” more clearly what is happening here,

Untitled-14
x^n for n=1,2,3….40.

Thus, the pointwise limit of the sequence of continuous functions is discontinuous at x=1. Remember our original assertion that this is true for all x in our domain D –well, that is not the case if we are talking about x in D=[0,1]. If we restrict our domain to (0,1) we “get away” with continuity.

In realit, pointwise limits are really not useful because the pointwise limit may not retain any of the useful properties that the original functions shared. We require something more useful and something more meaningful where the existence of it implies all our previous work involving pointwise limits. Nonetheless, our work was not a complete loss. In fact, our study of pointwise limits sets us up for Uniform Convergence.

Before jumping into uniform convergence, lets get some background.

UNIFORM LIMITS

In general, pointwise limits do NOT allow the interchange of limit operations (i.e. continuity at some point x_{0} in our domain \mathbb{D} we would have \lim\limits_{n \to \infty} f(x)=f(x_{0}) and this would absolutely require that \lim\limits_{x \to x_{0}}(\lim\limits_{n \to \infty} f_{n}(x))=\lim\limits_{n \to \infty}f(x_{0})=\lim\limits_{n \to \infty}(\lim\limits_{x \to x_{0}}f_{n}(x)) ). Uniform limits will often allow us to interchange the limit operations.

Definition 3:

Let {f_{n}} be a sequence of functions defined on a common domain D. We say that our sequence converges uniformly to a function {f} on D if for every \epsilon>0 there exists some N\in \mathbb{N} so that,

|f_{n} - f|<\epsilon for all n\neq \mathbb{N}.

We write, f_{n}\rightarrow {[unif]} on D.

In logical notation, we have: \forall {\epsilon}>0 \exists {N}\in\mathbb{N} : \forall{n}>{N} and \forall{x}\in\mathbb{D} there holds |f_{n}-f|<\epsilon

The major difference amid this definition and our the definition regarding piecewise limits is that N depends on \epsilon and NOT x. Essentially, for some value of n large, the value of the difference of our sequence of functions and the “actual” function is uniform.

Red denotes the f_n, black is f, and the dotted lines our our epsilon.
Red denotes the f_n, black is f, and the dotted lines are our epsilon bounds.

And important fact is that if f_{n}\rightarrow{f}{[unif]} then it also converges pointwise as well. This often allows us to start proofs by looking at the piecewise convergence and moving toward showing uniform convergence. This works because if the question at hand is asking for uniform convergence, one must assume that, at least, working first with what the question will eventually imply sets one up for a path to the solution -at the very least.

In the next post, we will cover the Cauchy Criterion, and, then, talk about uniform convergence.

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