Definition 3.6.1: Let S ⊆ ℝ. A function f : ℝ → ℝ is increasing
(resp. strictly increasing) on S if x < y with x, y ∈ S implies
f x ≤ f y (resp. f x < f y). It is decreasing (resp. strictly
decreasing) when the inequalities for f are reversed. A function that is
either increasing or decreasing on S is called monotone on S, and if it
is strictly increasing or strictly decreasing on S, it is called strictly
monotone on S.
Instances For
Equations
- monotoneOn f S = (increasingOn f S ∨ decreasingOn f S)
Instances For
Equations
- strictlyMonotoneOn f S = (strictlyIncreasingOn f S ∨ strictlyDecreasingOn f S)
Instances For
A technical lemma: if f is monotone on S, then the left limit along
S ∩ (-∞, c) converges to the supremum of the left slice when that slice is
bounded above.
Proposition 3.6.2: Let S ⊆ ℝ, c ∈ ℝ, f : ℝ → ℝ be increasing on S,
and g : ℝ → ℝ be decreasing on S. If c is a cluster point of
S ∩ (-∞, c), then lim_{x → c^-} f x = sup {f x | x ∈ S, x < c} and
lim_{x → c^-} g x = inf {g x | x ∈ S, x < c}. If c is a cluster point of
S ∩ (c, ∞), then lim_{x → c^+} f x = inf {f x | x ∈ S, x > c} and
lim_{x → c^+} g x = sup {g x | x ∈ S, x > c}. If ∞ is a cluster point of
S, then lim_{x → ∞} f x = sup {f x | x ∈ S} and
lim_{x → ∞} g x = inf {g x | x ∈ S}. If -∞ is a cluster point of S, then
lim_{x → -∞} f x = inf {f x | x ∈ S} and
lim_{x → -∞} g x = sup {g x | x ∈ S}.
If an interval I contains a point strictly to the left of c, then c
is a closure point of I ∩ (-∞, c).
If an interval I contains a point strictly to the right of c, then c
is a closure point of I ∩ (c, ∞).
Corollary 3.6.3: If I ⊆ ℝ is an interval and f : I → ℝ is monotone and
not constant, then the image f(I) is an interval if and only if f is
continuous.
Corollary 3.6.4: Let I ⊆ ℝ be an interval and f : I → ℝ be monotone. Then
f has at most countably many discontinuities.
Example 3.6.5: There exists a strictly increasing function f : [0, 1] → ℝ
that is bounded and has a discontinuity at each point 1 - 1/k for
k ∈ ℕ. In particular, it is monotone on a compact interval with
countably many discontinuities.
Proposition 3.6.6: If I ⊆ ℝ is an interval and f : I → ℝ is strictly
monotone, then the inverse f⁻¹ : f(I) → I is continuous.
Example 3.6.7: The piecewise function f : ℝ → ℝ given by
f x = x for x < 0 and f x = x + 1 for x ≥ 0 is not continuous at 0,
its image of ℝ is (-∞, 0) ∪ [1, ∞), and the inverse function
f⁻¹ : (-∞, 0) ∪ [1, ∞) → ℝ defined by y if y < 0 and y - 1 if y ≥ 1
is continuous.