Darboux's theorem #

In this file we prove that the derivative of a differentiable function on an interval takes all intermediate values. The proof is based on the Wikipedia page about this theorem.

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theorem exists_has_deriv_within_at_eq_of_gt_of_lt {a b : ℝ} {f f' : ℝ → ℝ} (hab : a ≤ b) (hf : ∀ (x : ℝ), x ∈ set.Icc a b → has_deriv_within_at f (f' x) (set.Icc a b) x) {m : ℝ} (hma : f' a < m) (hmb : m < f' b) :

m ∈ f' '' set.Icc a b

Darboux's theorem: if a ≤ b and f' a < m < f' b, then f' c = m for some c ∈ [a, b].

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theorem exists_has_deriv_within_at_eq_of_lt_of_gt {a b : ℝ} {f f' : ℝ → ℝ} (hab : a ≤ b) (hf : ∀ (x : ℝ), x ∈ set.Icc a b → has_deriv_within_at f (f' x) (set.Icc a b) x) {m : ℝ} (hma : m < f' a) (hmb : f' b < m) :

m ∈ f' '' set.Icc a b

Darboux's theorem: if a ≤ b and f' a > m > f' b, then f' c = m for some c ∈ [a, b].

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theorem convex_image_has_deriv_at {f f' : ℝ → ℝ} {s : set ℝ} (hs : convex ℝ s) (hf : ∀ (x : ℝ), x ∈ s → has_deriv_at f (f' x) x) :

convex ℝ (f' '' s)

Darboux's theorem: the image of a convex set under f' is a convex set.

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theorem deriv_forall_lt_or_forall_gt_of_forall_ne {f f' : ℝ → ℝ} {s : set ℝ} (hs : convex ℝ s) (hf : ∀ (x : ℝ), x ∈ s → has_deriv_at f (f' x) x) {m : ℝ} (hf' : ∀ (x : ℝ), x ∈ s → f' x ≠ m) :

(∀ (x : ℝ), x ∈ s → f' x < m) ∨ ∀ (x : ℝ), x ∈ s → m < f' x

If the derivative of a function is never equal to m, then either it is always greater than m, or it is always less than m.