# mathlib3documentation

analysis.calculus.darboux

# Darboux's theorem #

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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.

theorem exists_has_deriv_within_at_eq_of_gt_of_lt {a b : } {f f' : } (hab : a b) (hf : (x : ), x b (f' x) (set.Icc a b) x) {m : } (hma : f' a < m) (hmb : m < f' b) :
m f' '' b

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

theorem exists_has_deriv_within_at_eq_of_lt_of_gt {a b : } {f f' : } (hab : a b) (hf : (x : ), x b (f' x) (set.Icc a b) x) {m : } (hma : m < f' a) (hmb : f' b < m) :
m f' '' b

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

theorem set.ord_connected.image_has_deriv_within_at {f f' : } {s : set } (hs : s.ord_connected) (hf : (x : ), x s (f' x) s x) :

Darboux's theorem: the image of an ord_connected set under f' is an ord_connected set, has_deriv_within_at version.

theorem set.ord_connected.image_deriv_within {f : } {s : set } (hs : s.ord_connected) (hf : s) :

Darboux's theorem: the image of an ord_connected set under f' is an ord_connected set, deriv_within version.

theorem set.ord_connected.image_deriv {f : } {s : set } (hs : s.ord_connected) (hf : (x : ), x s ) :

Darboux's theorem: the image of an ord_connected set under f' is an ord_connected set, deriv version.

theorem convex.image_has_deriv_within_at {f f' : } {s : set } (hs : s) (hf : (x : ), x s (f' x) s x) :
(f' '' s)

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

theorem convex.image_deriv_within {f : } {s : set } (hs : s) (hf : s) :
s '' s)

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

theorem convex.image_deriv {f : } {s : set } (hs : s) (hf : (x : ), x s ) :
(deriv f '' s)

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

theorem exists_has_deriv_within_at_eq_of_ge_of_le {a b : } {f f' : } (hab : a b) (hf : (x : ), x b (f' x) (set.Icc a b) x) {m : } (hma : f' a m) (hmb : m f' b) :
m f' '' b

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

theorem exists_has_deriv_within_at_eq_of_le_of_ge {a b : } {f f' : } (hab : a b) (hf : (x : ), x b (f' x) (set.Icc a b) x) {m : } (hma : f' a m) (hmb : m f' b) :
m f' '' b

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

theorem has_deriv_within_at_forall_lt_or_forall_gt_of_forall_ne {f f' : } {s : set } (hs : s) (hf : (x : ), x s (f' x) s x) {m : } (hf' : (x : ), x s f' x m) :
( (x : ), x s f' x < m) (x : ), 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.