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For our final corollary, we first note a straightforward characterization: a set I ⊆ R is an interval if
a, b ∈ I and a < y < b =⇒ y ∈ I (∗)
Corollary 18.10 (Preservation of Intervals). Suppose f : U → V is continuous where U = dom f is
an interval (of positive length) and V = range f .
1. V is an interval or a point.
2. If f is strictly increasing (decreasing), then:
(a) V is an interval (it has positive length).
(b) f is injective (and thus bijective).
(c) The inverse function f
−1
: V → U is also continuous and strictly increasing (decreasing).
Example 18.11. In part 1, note that the interval V need not be of the
same type as U. For instance, if f (x) = 10x − x
2
, then f maps the open
interval U = ( 2, 9) to the half-open interval V = (9, 25].
The extreme value theorem, however, guarantees that if U is a closed
bounded interval, then V is also, for instance,
f
[2, 9]
= [9, 25]
Proof. 1. If V is not a point, then ∃a, b ∈ U such that f (a) < f (b). Let y ∈
f (a), f (b)
; IVT says
∃ξ between a and b such that y = f (ξ). That is, y ∈ range f . By (∗), V = range f is an interval.
2. (a,b) If f is strictly increasing, then ∀a, b ∈ U, a < b =⇒ f (a) < f (b). It follows that f is
injective and that V contains at least 2 points; by part 1 it has positive length.
(c) Let y
1
< y
2
where both lie in V, and define x
i
= f
−1
( y
i
) for i = 1, 2. Since f is increasing,
x
2
≤ x
1
=⇒ y
2
= f (x
2
) ≤ f (x
1
) = y
1
is a contradiction. Thus x
1
< x
2
and f
−1
is also strictly increasing.
It remains to show that f
−1
is continuous at b = f
−1
(a). Assume first that a is not an
endpoint of U. Given ϵ > 0 for which [a −ϵ, a + ϵ ] ⊆ U, let
δ := min
b − f (a − ϵ), f (a + ϵ) −b
and observe that
|
y − b
|
< δ =⇒ f (a −ϵ) − b < y −b < f (a + ϵ) − b =⇒ f (a − ϵ) < y < f (a + ϵ)
=⇒ a −ϵ < y < a + ϵ ( f strictly increasing)
=⇒
f
−1
( y) − a
< ϵ
If a is an endpoint of U, instead use [a − ϵ, a] ⊆ U or [a, a + ϵ] ⊆ U and only the corre-
sponding half of the expression δ.
75