Continuity: continued

Consider the theorem proved in the previous blog:

Theorem:

$f: \Re \rightarrow \Re$ is continuous at $x \in \Re$ if and only if $(f(x_{n}))_{n=1}^{\infty}$ converges to $f(x)$ whenever $(x_{n})_{n=1}^{\infty}$ converges to x, that is, $\lim_{n \rightarrow \infty}=f(\lim_{n \rightarrow \infty} x_{n})$ .

Observe that the above theorem states that not only does $(f(x_{n}))_{n=1}^{\infty}$ converge whenever $(x_{n})_{n=1}^{\infty}$ does, but also that $\lim f(x_{n})=f(\lim x_{n})$ for a continuous function. On the other hand, if f is not continuous it may happen that $\lim f(x_{n})$ exists, but does not equal $f(\lim x_{n})$ . This leads to another notion called limit of a function.

Definition. Suppose $f: \Re \rightarrow \Re$ . We say that $l$ is the limit of a function $f(x)$ as x tends to $x_{0}$ if for every $\varepsilon > 0$ , there is a $\delta > 0$ such that

$|f(x)-l|< \varepsilon$ for $0 < |x-x_{0}|< \delta$ .

In this case, we write $\lim_{x \rightarrow x_{0}} f(x)=l$ .

This is the same thing as saying that $f(x)$ can be brought as close to l as we please by bringing x sufficiently close to $x_{0}$ . But, we do not require the function to have the value $l$ at $x_{0}$ . If it has the value $l$ at $x_{0}$ . If it has the value $l$ at $x_{0}$ , then it is continuous at $x_{0}$ .

From the above discussion, we see that for $\lim_{x \rightarrow x_{0}} f(x)$ to exist it is not necessary for us to assume that f is defined at $x_{0}$ . We only need to know whether or not $f(x)$ is coming close to a definite real number when x is coming close to $x_{0}$ . Thus, for the limit of $f(x)$ (as x tends to $x_{0})$ to exist or not to exist, we need the function f to be defined for $\{ x: 0 < |x-x_{0}|< \delta\}$ for some $\delta >0$ x. This is the set of points in the interval $(x_{0}-\delta, x_{0}+\delta)$ from which $x_{0}$ has been deleted, that is, $\{ x:0 < |x-x_{0}|<\delta\}=(x_{0}-\delta, x_{0}) \bigcup (x_{0}, x_{0}+\delta)$ . When we want to look at a function over such a punctured neighbourhood of a point, we try to see what happens to the function when we come close to it, without actually reaching that point. This is not really artificial at all as many physical and mathematical exigencies force us to look at such situations. Take, the case of a particle moving in a straight line. With reference to a fixed point on the straight line, let $x(t)$ and $x(t_{0})$ be the positions of the particle at time t and $t_{0}$ respectively ( $t > t_{0}$ ). So its average velocity in the interval of time $[t_{0},t]$ is given by $\frac{x(t)-x(t_{0})}{t-t_{0}}=f(t)$ . Now the function f is defined for every t save $t_{0}$ . But, the instantaneous velocity should indeed e the $\lim_{t \rightarrow t_{0}}$ . In other words, as the interval of time $[t_{0},t]$ decreases, the average velocity should eventually stabilize to a certain number $v(t_{0})$ called the instantaneous velocity of the particle at the instant of time $t_{0}$ . If this happens, only then it is meaningful of talk of instantaneous velocity of the particle. See what we are doing. We have a function f defined for every real number except a particular real number $x_{0}$ . Then, we are trying to find out what happens to $f(t)$ as t comes closer and closer to $t_{0}$ . We shall see later many such situations of finding the limit of $f(t)$ as $t \rightarrow t_{0}$ , where $f(t)$ is defined for every t except $t=t_{0}$ .

Digress. The post office function.

When we want to mail a letter enclosed in an envelope, we usually go to the post master with it to tell us the denomination of the stamp to be affixed. The post master weighs the letter and tells us the postage according to the weight. Here, we have a definite postage for a definite weight, we don’t have different rates for the same weight (for same kind of mail like registered or ordinary or speed post). The rate chart with the post master, for a particular kind of mail, truly is a function whose domain is the set of positive real numbers representing the weight of the mail, and the range again consists of positive real numbers representing the postage. We write the chart as a function f such that $p=f(w)$ meaning p is the postage to mail a letter of weight w.

Let us look again at this post office function.

This function is clearly defined for every x. But what happens to $f(x)$ , for instance, when x comes close to 15? When we take $x=14.9, x=14.999$ we are, at each successive stage, coming closer to 15. Similarly, as we go through $x=15.1, x=15.01, x=15.001$ , we are coming closer to 15 as well. But, in the former case, we were approaching x through the values of x less than 15, or what is the same thing as approaching 15 from the left. In the latter case, we are approaching x through the values of x larger than x 15, or we are approaching from the right. It is clear that $\lim f(x)=2$ as x approaches 15 from the left, while $\lim f(x)=3$ , when x approaches 15 from the right. So we have the following definition:

Definition.

Let $f:\Re - \{ a\} \rightarrow \Re$ . We say that the left hand limit of $f(x)$ exists as x tends to a if there is a number l such that for every $\varepsilon > 0$ , we can find a $\delta > 0$ such that

$|f(x)-l|< \varepsilon$ for $a-\delta < x < a$ .

In such a case, we call l the left hand limit of $f(x)$ at a, and write $\lim_{x \rightarrow a_{-}} f(x)=l$ . Similarly, we say that $f(x)$ has right hand limit r as x tends to a if for all

$\varepsilon >0$ there exists $\delta >0$ such that

$|f(x)-r|< \varepsilon$ for $a < x < a+\delta$ .

In such a case, we call r the right hand limit of $f(x)$ as x tends to a and write $\lim_{x \rightarrow a_{+}}f(x)=r$ . It is clear that when the left hand limit exists, it is unique. So is the case with the right hand limit. Thus, we are led to the conclusion:

The limit of $f(x)$ exists as x tends to a if and only if both the left-hand and right hand limits of $f(x)$ exist as x approaches a, and they are equal.Moreover, if the common limit is equal to the value of the function at the point, then the function is continuous at that point

In the case of the post office function, the left hand and right hand limits exists at 15 but are not equal. So there is not the question of the limit existing much less the continuity of the function at 15.

More later,

Nalin Pithwa

Mathematics Hothouse