File: Mathlib/Topology/Order/LeftRightLim.lean

full_name: Function.leftLim
code: /-- Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and
let `a : α`. The limit strictly to the left of `f` at `a`, denoted with `leftLim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its left or the function has no left
limit, we use `f a` instead to guarantee a good behavior in most cases. -/
noncomputable def Function.leftLim (f : α → β) (a : α) : β 
goal: β

full_name: Function.rightLim
code: /-- Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and
let `a : α`. The limit strictly to the right of `f` at `a`, denoted with `rightLim f a`, is defined
by using the order topology on `α`. If `a` is isolated to its right or the function has no right
limit, , we use `f a` instead to guarantee a good behavior in most cases. -/
noncomputable def Function.rightLim (f : α → β) (a : α) : β 
goal: β

full_name: leftLim_eq_of_tendsto
code: theorem leftLim_eq_of_tendsto [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β]
    {f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Tendsto f (𝓝[<] a) (𝓝 y)) :
    leftLim f a = y 
goal: Function.leftLim f a = y

full_name: leftLim_eq_of_eq_bot
code: theorem leftLim_eq_of_eq_bot [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α}
    (h : 𝓝[<] a = ⊥) : leftLim f a = f a 
goal: Function.leftLim f a = f a

full_name: rightLim_eq_of_tendsto
code: theorem rightLim_eq_of_tendsto [TopologicalSpace α] [OrderTopology α] [T2Space β]
    {f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Tendsto f (𝓝[>] a) (𝓝 y)) :
    Function.rightLim f a = y 
goal: Function.rightLim f a = y

full_name: rightLim_eq_of_eq_bot
code: theorem rightLim_eq_of_eq_bot [TopologicalSpace α] [OrderTopology α] (f : α → β) {a : α}
    (h : 𝓝[>] a = ⊥) : rightLim f a = f a 
goal: Function.rightLim f a = f a

full_name: Monotone.leftLim_eq_sSup
code: theorem leftLim_eq_sSup [TopologicalSpace α] [OrderTopology α] (h : 𝓝[<] x ≠ ⊥) :
    leftLim f x = sSup (f '' Iio x) 
goal: Function.leftLim f x = sSup (f '' Set.Iio x)

full_name: Monotone.rightLim_eq_sInf
code: theorem rightLim_eq_sInf [TopologicalSpace α] [OrderTopology α] (h : 𝓝[>] x ≠ ⊥) :
    rightLim f x = sInf (f '' Ioi x) 
goal: Function.rightLim f x = sInf (f '' Set.Ioi x)

full_name: Monotone.leftLim_le
code: theorem leftLim_le (h : x ≤ y) : leftLim f x ≤ f y 
goal: Function.leftLim f x ≤ f y

full_name: Monotone.le_leftLim
code: theorem le_leftLim (h : x < y) : f x ≤ leftLim f y 
goal: f x ≤ Function.leftLim f y

full_name: Monotone.leftLim
code: @[mono]
protected theorem leftLim : Monotone (leftLim f) 
goal: Monotone (Function.leftLim f)

full_name: Monotone.le_rightLim
code: theorem le_rightLim (h : x ≤ y) : f x ≤ rightLim f y 
goal: f x ≤ Function.rightLim f y

full_name: Monotone.rightLim_le
code: theorem rightLim_le (h : x < y) : rightLim f x ≤ f y 
goal: Function.rightLim f x ≤ f y

full_name: Monotone.rightLim
code: @[mono]
protected theorem rightLim : Monotone (rightLim f) 
goal: Monotone (Function.rightLim f)

full_name: Monotone.leftLim_le_rightLim
code: theorem leftLim_le_rightLim (h : x ≤ y) : leftLim f x ≤ rightLim f y 
goal: Function.leftLim f x ≤ Function.rightLim f y

full_name: Monotone.rightLim_le_leftLim
code: theorem rightLim_le_leftLim (h : x < y) : rightLim f x ≤ leftLim f y 
goal: Function.rightLim f x ≤ Function.leftLim f y

full_name: Monotone.tendsto_leftLim
code: theorem tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) 
goal: Filter.Tendsto f (𝓝[<] x) (𝓝 (Function.leftLim f x))

full_name: Monotone.tendsto_leftLim_within
code: theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x) 
goal: Filter.Tendsto f (𝓝[<] x) (𝓝[≤] Function.leftLim f x)

full_name: Monotone.tendsto_rightLim
code: theorem tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) 
goal: Filter.Tendsto f (𝓝[>] x) (𝓝 (Function.rightLim f x))

full_name: Monotone.tendsto_rightLim_within
code: theorem tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≥] rightLim f x) 
goal: Filter.Tendsto f (𝓝[>] x) (𝓝[≥] Function.rightLim f x)

full_name: Monotone.continuousWithinAt_Iio_iff_leftLim_eq
code: /-- A monotone function is continuous to the left at a point if and only if its left limit
coincides with the value of the function. -/
theorem continuousWithinAt_Iio_iff_leftLim_eq :
    ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x 
goal: ContinuousWithinAt f (Set.Iio x) x ↔ Function.leftLim f x = f x

full_name: Monotone.continuousWithinAt_Ioi_iff_rightLim_eq
code: /-- A monotone function is continuous to the right at a point if and only if its right limit
coincides with the value of the function. -/
theorem continuousWithinAt_Ioi_iff_rightLim_eq :
    ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x 
goal: ContinuousWithinAt f (Set.Ioi x) x ↔ Function.rightLim f x = f x

full_name: Monotone.continuousAt_iff_leftLim_eq_rightLim
code: /-- A monotone function is continuous at a point if and only if its left and right limits
coincide. -/
theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x 
goal: ContinuousAt f x ↔ Function.leftLim f x = Function.rightLim f x

full_name: Monotone.countable_not_continuousWithinAt_Ioi
code: /-- In a second countable space, the set of points where a monotone function is not right-continuous
is at most countable. Superseded by `countable_not_continuousAt` which gives the two-sided
version. -/
theorem countable_not_continuousWithinAt_Ioi [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousWithinAt f (Ioi x) x } 
goal: {x | ¬ContinuousWithinAt f (Set.Ioi x) x}.Countable

full_name: Monotone.countable_not_continuousWithinAt_Iio
code: /-- In a second countable space, the set of points where a monotone function is not left-continuous
is at most countable. Superseded by `countable_not_continuousAt` which gives the two-sided
version. -/
theorem countable_not_continuousWithinAt_Iio [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousWithinAt f (Iio x) x } 
goal: {x | ¬ContinuousWithinAt f (Set.Iio x) x}.Countable

full_name: Monotone.countable_not_continuousAt
code: /-- In a second countable space, the set of points where a monotone function is not continuous
is at most countable. -/
theorem countable_not_continuousAt [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousAt f x } 
goal: {x | ¬ContinuousAt f x}.Countable

full_name: Antitone.le_leftLim
code: theorem le_leftLim (h : x ≤ y) : f y ≤ leftLim f x 
goal: f y ≤ Function.leftLim f x

full_name: Antitone.leftLim_le
code: theorem leftLim_le (h : x < y) : leftLim f y ≤ f x 
goal: Function.leftLim f y ≤ f x

full_name: Antitone.leftLim
code: @[mono]
protected theorem leftLim : Antitone (leftLim f) 
goal: Antitone (Function.leftLim f)

full_name: Antitone.rightLim_le
code: theorem rightLim_le (h : x ≤ y) : rightLim f y ≤ f x 
goal: Function.rightLim f y ≤ f x

full_name: Antitone.le_rightLim
code: theorem le_rightLim (h : x < y) : f y ≤ rightLim f x 
goal: f y ≤ Function.rightLim f x

full_name: Antitone.rightLim
code: @[mono]
protected theorem rightLim : Antitone (rightLim f) 
goal: Antitone (Function.rightLim f)

full_name: Antitone.rightLim_le_leftLim
code: theorem rightLim_le_leftLim (h : x ≤ y) : rightLim f y ≤ leftLim f x 
goal: Function.rightLim f y ≤ Function.leftLim f x

full_name: Antitone.leftLim_le_rightLim
code: theorem leftLim_le_rightLim (h : x < y) : leftLim f y ≤ rightLim f x 
goal: Function.leftLim f y ≤ Function.rightLim f x

full_name: Antitone.tendsto_leftLim
code: theorem tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) 
goal: Filter.Tendsto f (𝓝[<] x) (𝓝 (Function.leftLim f x))

full_name: Antitone.tendsto_leftLim_within
code: theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≥] leftLim f x) 
goal: Filter.Tendsto f (𝓝[<] x) (𝓝[≥] Function.leftLim f x)

full_name: Antitone.tendsto_rightLim
code: theorem tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) 
goal: Filter.Tendsto f (𝓝[>] x) (𝓝 (Function.rightLim f x))

full_name: Antitone.tendsto_rightLim_within
code: theorem tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≤] rightLim f x) 
goal: Filter.Tendsto f (𝓝[>] x) (𝓝[≤] Function.rightLim f x)

full_name: Antitone.continuousWithinAt_Iio_iff_leftLim_eq
code: /-- An antitone function is continuous to the left at a point if and only if its left limit
coincides with the value of the function. -/
theorem continuousWithinAt_Iio_iff_leftLim_eq :
    ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x 
goal: ContinuousWithinAt f (Set.Iio x) x ↔ Function.leftLim f x = f x

full_name: Antitone.continuousWithinAt_Ioi_iff_rightLim_eq
code: /-- An antitone function is continuous to the right at a point if and only if its right limit
coincides with the value of the function. -/
theorem continuousWithinAt_Ioi_iff_rightLim_eq :
    ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x 
goal: ContinuousWithinAt f (Set.Ioi x) x ↔ Function.rightLim f x = f x

full_name: Antitone.continuousAt_iff_leftLim_eq_rightLim
code: /-- An antitone function is continuous at a point if and only if its left and right limits
coincide. -/
theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x 
goal: ContinuousAt f x ↔ Function.leftLim f x = Function.rightLim f x

full_name: Antitone.countable_not_continuousAt
code: /-- In a second countable space, the set of points where an antitone function is not continuous
is at most countable. -/
theorem countable_not_continuousAt [SecondCountableTopology β] :
    Set.Countable { x | ¬ContinuousAt f x } 
goal: {x | ¬ContinuousAt f x}.Countable