258 CHAPTER 10. NORMED LINEAR SPACES

Definition 10.1.44 Let fn : X → Y where (X ,d) ,(Y,ρ) are two metric spaces. Then { fn}is said to converge pointwise to a function f : X → Y if for every x ∈ X ,

limn→∞

fn (x) = f (x)

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

supx∈X

ρ ( fn (x) , f (x))< ε

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

Example 10.1.45 Let fn (x) = xn on the metric space [0,1] . Then this function convergespointwise to

f (x) =

{0 on [0,1)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 thougheach function in the sequence is. The nice thing about uniform convergence is that it takescontinuity of the functions in the sequence and imparts it to the target function. It does thisfor both continuity at a single point and uniform continuity. Thus uniform convergence isa very superior thing.

Theorem 10.1.46 Let fn : X → Y where (X ,d) ,(Y,ρ) are two metric spaces and supposeeach fn is continuous at x ∈ X and also that fn converges uniformly to f on X. Then f isalso continuous at x. In addition to this, if each fn is uniformly continuous on X , then thesame is true for f .

Proof: Let ε > 0 be given. Then

ρ ( f (x) , f (x̂))≤ ρ ( f (x) , fn (x))+ρ ( fn (x) , fn (x̂))+ρ ( fn (x̂) , f (x̂))

By uniform convergence, there exists N such that ρ ( f (x) , fn (x)), ρ ( fn (x̂) , f (x̂)) are eachless than ε/3 provided n≥ N. Thus picking such an n,

ρ ( f (x) , f (x̂))≤ 2ε

3+ρ ( fn (x) , fn (x̂))

Now from the continuity of fn, there exists δ > 0 such that if d (x, x̂)< δ , then

ρ ( fn (x) , fn (x̂))< ε/3.

Hence, if d (x, x̂)< δ , then

ρ ( f (x) , f (x̂))≤ 2ε

3+ρ ( fn (x) , fn (x̂))<

2ε

3+

ε

3= ε

Hence, f is continuous at x.Next consider uniform continuity. It follows from the uniform convergence that if x, x̂

are any two points of X , then if n≥ N, then, picking such an n,

ρ ( f (x) , f (x̂))≤ 2ε

3+ρ ( fn (x) , fn (x̂))

By uniform continuity of fn there exists δ such that if d (x, x̂) < δ , then the term on theright in the above is less than ε/3. Hence if d (x, x̂)< δ , then ρ ( f (x) , f (x̂))< ε and so fis uniformly continuous as claimed. ■