3.3 Continuity  (Page 13/2)

Next, we come to the definition of continuity. Unlike the preceding discussion, which can beviewed as being related primarily to the algebraic properties of functions, this one is an analytic notion.

Let $S$ and $T$ be sets of complex numbers, and let $f:S\to T.$ Then $f$ is said to be continuous at a point $c$ of $S$ if for every positive $ϵ,$ there exists a positive $\delta$ such that if $x\in S$ satisfies $|x-c|<\delta ,$ then $\left|f\right(x)-f(c\left)\right|<ϵ.$ The function $f$ is called continuous on S if it is continuous at every point $c$ of $S.$

If the domain $S$ of $f$ consists of real numbers, then the function $f$ is called right continuous at c if for every $ϵ>0$ there exists a $\delta >0$ such that $\left|f\right(x)-f(c\left)\right|<ϵ$ whenever $x\in S$ and $0\le x-c<\delta ,$ and is called left continuous at $c$ if for every $ϵ>0$ there exists a $\delta >0$ such that $\left|f\right(x)-f(c\left)\right|<ϵ$ whenever $x\in S$ and $0\ge x-c>-\delta .$

REMARK If $f$ is continuous at a point $c,$ then the positive number $\delta$ of the preceding definition is not unique (any smaller number would work as well), but it does depend both on the number $ϵ$ and on the point $c.$ Sometimes we will write $\delta (ϵ,c)$ to make this dependence explicit. Later, we will introduce a notion of uniform continuityin which $\delta$ only depends on the number $ϵ$ and not on the particular point $c.$

The next theorem indicates the interaction between the algebraic properties of functions and continuity.

Let $S$ and $T$ be subsets of $C,$ let $f$ and $g$ be functions from $S$ into $T,$ and suppose that $f$ and $g$ are both continuous at a point $c$ of $S.$ Then

1. There exists a $\delta >0$ and a positive number $M$ such that if $|y-c|<\delta$ and $y\in S$ then $\left|f\right(y\left)\right|\le M.$ That is, if $f$ is continuous at $c,$ then it is bounded near $c.$
2. $f+g$ is continuous at $c.$
3. $fg$ is continuous at $c.$
4. $\left|f\right|$ is continuous at $c.$
5. If $g\left(c\right)\ne 0,$ then $f/g$ is continuous at $c.$
6. If $f$ is a complex-valued function, and $u$ and $v$ are the real and imaginary parts of $f,$ then $f$ is continuous at $c$ if and only if $u$ and $v$ are continuous at $c.$

We prove parts (1) and (5), and leave the remaining parts to the exercise that follows.

To see part (1), let $ϵ=1.$ Then, since $f$ is continuous at $c,$ there exists a $\delta >0$ such that if $|y-c|<\delta$ and $y\in S$ then $\left|f\right(y)-f(c\left)\right|<1.$ Since $|z-w|\ge \left|\right|z|-|w\left|\right|$ for any two complex numbers $z$ and $w$ (backwards Triangle Inequality), it then follows that $\left|\right|f\left(y\right)|-|f\left(c\right)\left|\right|<1,$ from which it follows that if $|y-c|<\delta$ then $\left|f\right(y\left)\right|<\left|f\right(c\left)\right|+1.$ Hence, setting $M=\left|f\right(c\left)\right|+1,$ we have that if $|y-c|<\delta$ and $y\in S,$ then $\left|f\right(y\left)\right|\le M$ as desired.

To prove part (5), we first make use of part 1. Let ${\delta }_1,M_1$ and ${\delta }_2,M_2$ be chosen so that if $|y-c|<{\delta }_1$ and $y\in S$ then

and if $|y-c|<{\delta }_2$ and $y\in S$ then

Next, let ${ϵ}^{\text{'}}$ be the positive number $\left|g\right(c\left)\right|/2.$ Then, there exists a ${\delta }^{\text{'}}>0$ such that if $|y-c|<{\delta }^{\text{'}}$ and $y\in S$ then $|g\left(y\right)-g\left(c\right)|<{ϵ}^{\text{'}}=\left|g\left(c\right)\right|/2.$ It then follows from the backwards triangle inequality that

Now, to finish the proof of part (5), let $ϵ>0$ be given. If $|y-c|<min({\delta }_1,{\delta }_2,{\delta }^{\text{'}})$ and $y\in S,$ then from Inequalities (3.1), (3.2), and (3.3) we obtain