Chapter 7

The DerivativeSome functions have derivatives and some don’t. Some have derivatives at some pointsand not at others. This chapter is on the derivative. Functions which have derivatives aresomehow better than those which don’t. To begin with it is necessary to discuss the conceptof a limit of a function. This is a harder concept than continuity and it is also harder than theconcept of the limit of a sequence or series although that is similar. One cannot make anyrational sense of the concept of derivative without an understanding of limits of a function.This is the main reason for considering the notion of limit.

7.1 Limit of a FunctionFor now, I will continue considering functions defined on a subset D( f ) of Fp having valuesin Fq.

Definition 7.1.1 Let x be a limit point of D( f ) . Then limy→x f (y) = L means thatfor every ε > 0 there is δ > 0 such that whenever 0 < ∥y− x∥ < δ , with y ∈ D( f ) , then∥ f (y)−L∥< ε .

Definition 7.1.2 Let x be a limit point of D( f ) and let f have values in R. Thenlimy→x f (y)=∞ means that for l, there is δ > 0 such that if 0< ∥y− x∥< δ , with y∈D( f ) ,then f (y)> l. A similar definition holds to define limy→x f (y) =−∞. If D( f ) contains aninterval (a,∞) , then limx→∞ f (x) = L ∈ Fp means that for every ε > 0 there exists l suchthat if x > l, then ∥ f (x)−L∥< ε . limx→−∞ f (x) = L is defined similarly.

I will leave for the reader the appropriate definition in the case that x→∞ and f (x)→∞

and other such cases.

Theorem 7.1.3 If limy→x f (y) = L and limy→x f (y) = L1, then L = L1. Uniquenessalso holds for one sided limits and for limits as x→ ∞ or x→−∞.

Proof: Let ε > 0 be given. There exists δ > 0 such that if 0 < |y− x| < δ , then| f (y)−L| < ε, | f (y)−L1| < ε. Therefore, for such y,(It exists because x is a limit point)|L−L1| ≤ |L− f (y)|+ | f (y)−L1| < ε + ε = 2ε. Since ε > 0 was arbitrary, this showsL = L1. The argument is exactly the same in the case of one sided limits. You simply needto have some y close enough to x on one side and in the case of limits at ±∞, you use ysuch that |y| is sufficiently large.

In the special case that f is defined near a point x, we sometimes speak of left and rightlimits by restricting the domain to be either those y < x or those y > x. When this is done,one writes limy→x+ f (y) or limy→x− f (y).

The first thing to do is to give an easier to use description in terms of sequences.

Proposition 7.1.4 Let x be a limit point of D( f ) . Then limy→x f (y) = L ∈ Fq if andonly if whenever xn→ x for each xn ̸= x, the xn distinct points, it follows that f (xn)→ L.

Proof: ⇒ Let xn → x where no xn equals x. Let ε > 0 be given. By assumption,| f (y)−L| < ε whenever 0 < |y− x| < δ for some δ . However, for all n large enough,0 < |xn− x|< δ and so | f (xn)−L|< ε. Hence f (xn)→ L.⇐ Suppose the condition on the sequences holds. If the condition for the limit does

not hold, then there exists ε > 0 such that no matter how small δ , there will be 0 <

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