114 CHAPTER 6. CONTINUOUS FUNCTIONS

12. Suppose K ⊆ Fp is a compact set and f : K→ Fq is continuous and one to one. Showthat f−1 : f (K)→ K is continuous.

6.9 Sequences and Series of FunctionsWhen you understand sequences and series of numbers it is easy to consider sequences andseries of functions.

Definition 6.9.1 A sequence of functions is a map defined on N or some set of inte-gers larger than or equal to a given integer, m which has values which are functions. It iswritten in the form { fn}∞

n=m where fn is a function. It is assumed also that the domain ofall these functions is the same.

In the above, where do the functions have values? Are they real valued functions? Arethey complex valued functions? Are they functions which have values in Rn? It turns outit does not matter very much and the same definition holds. However, if you like, you canthink of them as having values in F. This is the main case of interest here.

Example 6.9.2 Suppose fn (x) = xn for x ∈ [0,1] . Here is a graph of the functions f (x) =x,x2,x3,x4,x5.

x

y

Definition 6.9.3 Let { fn} be a sequence of functions. Then the sequence convergespointwise to a function f if for all x ∈ D, the domain of the functions in the sequence,

f (x) = limn→∞

fn (x)

This is always the definition regardless of where the fn have their values.

Thus you consider for each x ∈ D the sequence { fn (x)} and if this sequence convergesfor each x ∈ D, the thing it converges to is called f (x).

Example 6.9.4 In Example 6.9.2 find limn→∞ fn.

For x∈ [0,1), limn→∞ xn = fn (x) = 0. At x = 1, fn (1) = 1 for all n so limn→∞ fn (1) = 1.Therefore, this sequence of functions converges pointwise to the function f (x) given byf (x) = 0 if 0 ≤ x < 1 and f (1) = 1. However, given small ε > 0, and n, there is alwayssome x such that | f (x)− fn (x)|> ε . Just pick x less than 1 but close to 1. Then f (x) = 0but fn (x) will be close to 1.

Pointwise convergence is a very inferior thing but sometimes it is all you can get. It’sundesirability is illustrated by Example 6.9.4. The limit function is not continuous althougheach fn is continuous. Now here is another example of a sequence of functions.

Example 6.9.5 Let fn (x) = 1n sin

(n2x).