710 CHAPTER 22. THE DERIVATIVE

Theorem 22.1.6 For f : D( f )→W and x ∈ D( f ) a limit point of D( f ), f is continuousat x if and only if

limy→x

f (y) = f (x) .

Proof: First suppose f is continuous at x a limit point of D( f ). Then for every ε > 0there exists δ > 0 such that if ∥x− y∥ < δ and y ∈ D( f ), then | f (x)− f (y)| < ε . Inparticular, this holds if 0 < ∥x− y∥ < δ and this is just the definition of the limit. Hencef (x) = limy→x f (y).

Next suppose x is a limit point of D( f ) and limy→x f (y) = f (x). This means that if ε >0 there exists δ > 0 such that for 0 < ∥x− y∥< δ and y ∈D( f ), it follows | f (y)− f (x)|<ε . However, if y = x, then | f (y)− f (x)| = | f (x)− f (x)| = 0 and so whenever y ∈ D( f )and ∥x− y∥< δ , it follows | f (x)− f (y)|< ε , showing f is continuous at x.

Example 22.1.7 Find lim(x,y)→(3,1)

(x2−9x−3 ,y

).

It is clear that lim(x,y)→(3,1)x2−9x−3 = 6 and lim(x,y)→(3,1) y= 1. Therefore, this limit equals

(6,1).

Example 22.1.8 Find lim(x,y)→(0,0)xy

x2+y2 .

First of all, observe the domain of the function is R2 \{(0,0)}, every point in R2 exceptthe origin. Therefore, (0,0) is a limit point of the domain of the function so it might makesense to take a limit. However, just as in the case of a function of one variable, the limit maynot exist. In fact, this is the case here. To see this, take points on the line y = 0. At thesepoints, the value of the function equals 0. Now consider points on the line y = x where thevalue of the function equals 1/2. Since, arbitrarily close to (0,0), there are points wherethe function equals 1/2 and points where the function has the value 0, it follows there canbe no limit. Just take ε = 1/10 for example. You cannot be within 1/10 of 1/2 and alsowithin 1/10 of 0 at the same time.

Note it is necessary to rely on the definition of the limit much more than in the case ofa function of one variable and there are no easy ways to do limit problems for functions ofmore than one variable. It is what it is and you will not deal with these concepts withoutsuffering and anguish.

22.2 Basic DefinitionsThe concept of derivative generalizes right away to functions defined on a normed linearspace. However, no attempt will be made to consider derivatives from one side or another.This is because there isn’t a well defined side. However, it is certainly the case that thereare more general notions which include such things. I will present a fairly general notionof the derivative of a function which is defined on a normed vector space which has valuesin a normed vector space.

In what follows, X ,Y will denote normed vector spaces. Recall that L (X ,Y ) willdenote the bounded linear transformations from X to Y .

Let U be an open set in X , and let f : U → Y be a function.