Next is to consider the meaning of convergence of sequences of functions. There are two main ways of
convergence of interest here, pointwise and uniform convergence.

Definition 11.1.43Let f_{n} : X → Y where

(X,d)

,

(Y,ρ)

are two metric spaces. Then

{fn}

is said toconverge pointwise to a function f : X → Y if for every x ∈ X,

nli→m∞fn (x ) = f (x)

{fn}

is said to converge uniformly if for all ε > 0, there exists N such that if n ≥ N, then

sup ρ(fn(x),f (x)) < ε
x∈X

Here is a well known example illustrating the difference between pointwise and uniform
convergence.

Example 11.1.44Let f_{n}

(x)

= x^{n}on the metric space

[0,1]

. Then this function converges pointwiseto

{
0 on [0,1)
f (x) = 1 at 1

but it does not converge uniformly on this interval to f.

Note how the target function f in the above example is not continuous even though each
function in the sequence is. The nice thing about uniform convergence is that it takes continuity
of the functions in the sequence and imparts it to the target function. It does this for both
continuity at a single point and uniform continuity. Thus uniform convergence is a very superior
thing.

Theorem 11.1.45Let f_{n} : X → Y where

(X,d)

,

(Y,ρ)

are two metric spaces and suppose each f_{n}is continuous at x ∈ X and also that f_{n}convergesuniformly to f on X. Then f is also continuousat x. In addition to this, if each f_{n}is uniformly continuous on X, then the same is true for f.

Proof:Let ε > 0 be given. Then

ρ(f (x),f (ˆx)) ≤ ρ (f (x ),f (x ))+ ρ (f (x ),f (xˆ))+ ρ (f (xˆ),f (ˆx))
n n n n

By uniform convergence, there exists N such that both ρ

(f (x),fn(x))

and ρ

(fn(ˆx),f (ˆx))

are less than
ε∕3 provided n ≥ N. Thus picking such an n,

ρ(f (x),f (ˆx)) ≤ 2ε + ρ(fn(x),fn(ˆx))
3

Now from the continuity of f_{n}, there exists δ > 0 such that if d