File: Mathlib/Analysis/Calculus/Darboux.lean

full_name: exists_hasDerivWithinAt_eq_of_gt_of_lt
code: /-- **Darboux's theorem**: if `a ≤ b` and `f' a < m < f' b`, then `f' c = m` for some
`c ∈ (a, b)`. -/
theorem exists_hasDerivWithinAt_eq_of_gt_of_lt (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a < m)
    (hmb : m < f' b) : m ∈ f' '' Ioo a b 
goal: m ∈ f' '' Set.Ioo a b

full_name: exists_hasDerivWithinAt_eq_of_lt_of_gt
code: /-- **Darboux's theorem**: if `a ≤ b` and `f' b < m < f' a`, then `f' c = m` for some `c ∈ (a, b)`.
-/
theorem exists_hasDerivWithinAt_eq_of_lt_of_gt (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : m < f' a)
    (hmb : f' b < m) : m ∈ f' '' Ioo a b 
goal: m ∈ f' '' Set.Ioo a b

full_name: Set.OrdConnected.image_hasDerivWithinAt
code: /-- **Darboux's theorem**: the image of a `Set.OrdConnected` set under `f'` is a `Set.OrdConnected`
set, `HasDerivWithinAt` version. -/
theorem Set.OrdConnected.image_hasDerivWithinAt {s : Set ℝ} (hs : OrdConnected s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : OrdConnected (f' '' s) 
goal: (f' '' s).OrdConnected

full_name: Set.OrdConnected.image_derivWithin
code: /-- **Darboux's theorem**: the image of a `Set.OrdConnected` set under `f'` is a `Set.OrdConnected`
set, `derivWithin` version. -/
theorem Set.OrdConnected.image_derivWithin {s : Set ℝ} (hs : OrdConnected s)
    (hf : DifferentiableOn ℝ f s) : OrdConnected (derivWithin f s '' s) 
goal: (derivWithin f s '' s).OrdConnected

full_name: Set.OrdConnected.image_deriv
code: /-- **Darboux's theorem**: the image of a `Set.OrdConnected` set under `f'` is a `Set.OrdConnected`
set, `deriv` version. -/
theorem Set.OrdConnected.image_deriv {s : Set ℝ} (hs : OrdConnected s)
    (hf : ∀ x ∈ s, DifferentiableAt ℝ f x) : OrdConnected (deriv f '' s) 
goal: (deriv f '' s).OrdConnected

full_name: Convex.image_hasDerivWithinAt
code: /-- **Darboux's theorem**: the image of a convex set under `f'` is a convex set,
`HasDerivWithinAt` version. -/
theorem Convex.image_hasDerivWithinAt {s : Set ℝ} (hs : Convex ℝ s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : Convex ℝ (f' '' s) 
goal: Convex ℝ (f' '' s)

full_name: Convex.image_derivWithin
code: /-- **Darboux's theorem**: the image of a convex set under `f'` is a convex set,
`derivWithin` version. -/
theorem Convex.image_derivWithin {s : Set ℝ} (hs : Convex ℝ s) (hf : DifferentiableOn ℝ f s) :
    Convex ℝ (derivWithin f s '' s) 
goal: Convex ℝ (derivWithin f s '' s)

full_name: Convex.image_deriv
code: /-- **Darboux's theorem**: the image of a convex set under `f'` is a convex set,
`deriv` version. -/
theorem Convex.image_deriv {s : Set ℝ} (hs : Convex ℝ s) (hf : ∀ x ∈ s, DifferentiableAt ℝ f x) :
    Convex ℝ (deriv f '' s) 
goal: Convex ℝ (deriv f '' s)

full_name: exists_hasDerivWithinAt_eq_of_ge_of_le
code: /-- **Darboux's theorem**: if `a ≤ b` and `f' a ≤ m ≤ f' b`, then `f' c = m` for some
`c ∈ [a, b]`. -/
theorem exists_hasDerivWithinAt_eq_of_ge_of_le (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a ≤ m)
    (hmb : m ≤ f' b) : m ∈ f' '' Icc a b 
goal: m ∈ f' '' Set.Icc a b

full_name: exists_hasDerivWithinAt_eq_of_le_of_ge
code: /-- **Darboux's theorem**: if `a ≤ b` and `f' b ≤ m ≤ f' a`, then `f' c = m` for some
`c ∈ [a, b]`. -/
theorem exists_hasDerivWithinAt_eq_of_le_of_ge (hab : a ≤ b)
    (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a ≤ m)
    (hmb : m ≤ f' b) : m ∈ f' '' Icc a b 
goal: m ∈ f' '' Set.Icc a b

full_name: hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne
code: /-- 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`. -/
theorem hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne {s : Set ℝ} (hs : Convex ℝ s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) {m : ℝ} (hf' : ∀ x ∈ s, f' x ≠ m) :
    (∀ x ∈ s, f' x < m) ∨ ∀ x ∈ s, m < f' x 
goal: (∀ x ∈ s, f' x < m) ∨ ∀ x ∈ s, m < f' x