Measure, Integration & Real Analysis, 2021a

150 Chapter 6 Banach Spaces
The definition of continuity that follows uses the same pattern as the definition for
a function from a subset of R to R.
6.10 Definition continuous
Suppose (V, d V ) and (W, d W ) are metric spaces and T : V → W is a function.
• For f ∈ V, the function T is called continuous at f if for every ε > 0, there
exists δ > 0 such that
( )
d W T( f ), T(g) < ε
for all g ∈ V with d V ( f , g) < δ.
• The function T is called continuous if T is continuous at f for every f ∈ V.
The next result gives equivalent conditions for continuity. Recall that T −1 (E) is
called the inverse image of E and is defined to be { f ∈ V : T( f ) ∈ E}. Thus the
equivalence of the (a) and (c) below could be restated as saying that a function is
continuous if and only if the inverse image of every open set is open. The equivalence
of the (a) and (d) below could be restated as saying that a function is continuous if
and only if the inverse image of every closed set is closed.
6.11 equivalent conditions for continuity
Suppose V and W are metric spaces and T : V → W is a function. Then the
following are equivalent:
(a) T is continuous.
(b) lim f k = f in V implies lim T( f k )=T( f ) in W.
k→∞ k→∞
(c) T −1 (G) is an open subset of V for every open set G ⊂ W.
(d) T −1 (F) is a closed subset of V for every closed set F ⊂ W.
Proof We first prove that (b) implies (d). Suppose (b) holds. Suppose F is a closed
subset of W. We need to prove that T −1 (F) is closed. To do this, suppose f 1 , f 2 ,...
is a sequence in T −1 (F) and lim k→∞ f k = f for some f ∈ V. Because (b) holds, we
know that lim k→∞ T( f k )=T( f ). Because f k ∈ T −1 (F) for each k ∈ Z + , we know
that T( f k ) ∈ F for each k ∈ Z + . Because F is closed, this implies that T( f ) ∈ F.
Thus f ∈ T −1 (F), which implies that T −1 (F) is closed [by 6.9(e)], completing the
proof that (b) implies (d).
The proof that (c) and (d) are equivalent follows from the equation
T −1 (W \ E) =V \ T −1 (E)
for every E ⊂ W and the fact that a set is open if and only if its complement (in the
appropriate metric space) is closed.
The proof of the remaining parts of this result are left as an exercise that should
help strengthen your understanding of these concepts.
Measure, Integration & Real Analysis, by Sheldon Axler