Download CONTINUITY IN METRIC SPACES

Notes 4
CONTINUITY IN METRIC SPACES
Let (X, d) and (Y, ρ) be metric spaces, and let f : X → Y be a function.
Definition. A function f is called continuous at x ∈ X if for every ε > 0 there
exists δ > 0 such that ρ( f (y), f (x)) < ε whenever d(y, x) < δ.
If f is continuous for all x ∈ X, we say that f is continuous on X.
Remark. Our previous definitions of continuity for functions R → R or C → C
are special cases of the definition above.
Theorem. A function f : X → Y is continuous at x ∈ X iff the sequence { f (xn )}
converges to f (x) for any sequence {xn } converging to x.
Theorem. Let X, Y , Z be metric spaces. If f : X → Y is continuous at x ∈ X and
g : Y → Z is continuous at f (x) ∈ Y , then g ◦ f : X → Z is continuos at x.
Example.
(a) The function f : R → R given by
(
for
x sin 1x
f (x) =
0
for
x 6= 0 ,
x = 0,
is continuous for all x ∈ R.
(b) The function f : R → R given by
(
sin 1x
f (x) =
0
for
for
x 6= 0 ,
x = 0,
is not continuous at x = 0.
(c) The function f : R2 → R given by
( 2xy
f (x, y) =
x2 +y2
for
(x, y) 6= (0, 0) ,
0
for
(x, y) = (0, 0) ,
is not continuous at (x, y) = (0, 0).
(d) The function f : R2 → R given by
( xy
√
x2 +y2
f (x, y) =
0
is continuous at (x, y) = (0, 0).
1
for
(x, y) 6= (0, 0) ,
for
(x, y) = (0, 0) ,
(e) The function T : C[0, 1] → C[0, 1] given by [T ( f )](x) = f 2 (x), is continuous
at f for all f ∈ C[0, 1].
Continuity and open sets.
Definition. Suppose A ⊆ X, B ⊆ Y , and f : X → Y . The set
f (A) = {y ∈ Y : y = f (x) for some x ∈ X} = { f (x) : x ∈ A}
is called the image of A and the set
f −1 (B) = {x ∈ X : f (x) ∈ B}
is called the pre-image of B.
(Thus x ∈ f −1 (B) iff f (x) ∈ B.)
Theorem. Let (X, d) and (Y, ρ) be metric spaces and suppose f : X → Y . Then f
is continuous on X iff f −1 (O) is an open set in X whenever O is an open set in Y .
As a formal logical statements, this theorem can be written in the following form:
( f is continuous on X) ⇐⇒ ((O open in Y ) =⇒ ( f −1 (O) open in X)) .
Sequences of continuous functions.
Suppose that { fn } is a sequence of continuous functions converging to a function
f.
If convergence is pointwise, then the limit f is not necessarily continuous, consider, for example, fn (x) = xn for x ∈ [0, 1].
However, in the case of uniform convergence we have
Theorem. Let X be a real interval, and suppose { fn } is a sequence of continuous
functions in B(X) such that { fn } converges uniformly to f . Then f is continuous
on X.
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