448 CHAPTER 21. FUNCTIONS OF MANY VARIABLES

Definition 21.2.4 Let f : D(f) ⊆ Rp → Rq where q, p ≥ 1 be a function and letx be a limit point of D(f). Then limy→xf (y) = L if and only if the following condi-tion holds. For all ε > 0 there exists δ > 0 such that if 0 < |y−x| < δ and y ∈ D(f)then,|L−f (y)|< ε.

The condition that x must be a limit point of D(f) if you are to take a limit at x is whatmakes the limit well defined.

Proposition 21.2.5 Let f : D(f) ⊆ Rp → Rq where q, p ≥ 1 be a function and let xbe a limit point of D(f). Then if limy→xf (y) exists, it must be unique.

Proof: Suppose limy→xf (y) =L1 and limy→xf (y) =L2. Then for ε > 0 given, letδ i > 0 correspond to Li in the definition of the limit and let δ = min(δ 1,δ 2). Since x is alimit point, there exists y ∈ B(x,δ )∩D(f). Therefore,

|L1 −L2| ≤ |L1 −f (y)|+ |f (y)−L2|< ε + ε = 2ε.

Since ε > 0 is arbitrary, this shows L1 =L2.The following theorem summarized many important interactions involving continuity.

Most of this theorem has been proved in Theorem 15.8.6 on Page 318.

Theorem 21.2.6 Suppose x is a limit point of D(f) and

limy→x

f (y) =L, limy→x

g (y) =K

where K and L are vectors in Rp for p ≥ 1. Then if a, b ∈ R,

limy→x

af (y)+bg (y) = aL+bK, (21.1)

limy→x

f ·g (y) =L ·K (21.2)

Also, if h is a continuous function defined near L, then

limy→x

h◦f (y) = h(L) . (21.3)

For a vector valued function

f (y) = ( f1 (y) , · · · , fq (y))T ,

limy→xf (y) =L= (L1 · · · ,Lk)T if and only if

limy→x

fk (y) = Lk (21.4)

for each k = 1, · · · , p.In the case where f and g have values in R3

limy→x

f (y)×g (y) =L×K. (21.5)

Also recall Theorem 15.8.7 on Page 319.

Theorem 21.2.7 For f : D(f)→ Rq and x ∈ D(f) such that x is a limit point ofD(f), it follows f is continuous at x if and only if limy→xf (y) = f (x).