File: Mathlib/Analysis/Calculus/Monotone.lean

full_name: tendsto_apply_add_mul_sq_div_sub
code: /-- If `(f y - f x) / (y - x)` converges to a limit as `y` tends to `x`, then the same goes if
`y` is shifted a little bit, i.e., `f (y + (y-x)^2) - f x) / (y - x)` converges to the same limit.
This lemma contains a slightly more general version of this statement (where one considers
convergence along some subfilter, typically `𝓝[<] x` or `𝓝[>] x`) tailored to the application
to almost everywhere differentiability of monotone functions. -/
theorem tendsto_apply_add_mul_sq_div_sub {f : ℝ → ℝ} {x a c d : ℝ} {l : Filter ℝ} (hl : l ≤ 𝓝[≠] x)
    (hf : Tendsto (fun y => (f y - d) / (y - x)) l (𝓝 a))
    (h' : Tendsto (fun y => y + c * (y - x) ^ 2) l l) :
    Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a) 
goal: Filter.Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a)

full_name: StieltjesFunction.ae_hasDerivAt
code: /-- A Stieltjes function is almost everywhere differentiable, with derivative equal to the
Radon-Nikodym derivative of the associated Stieltjes measure with respect to Lebesgue. -/
theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
    ∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x 
goal: ∀ᵐ (x : ℝ), HasDerivAt (↑f) (f.measure.rnDeriv MeasureTheory.volume x).toReal x

full_name: Monotone.ae_hasDerivAt
code: /-- A monotone function is almost everywhere differentiable, with derivative equal to the
Radon-Nikodym derivative of the associated Stieltjes measure with respect to Lebesgue. -/
theorem Monotone.ae_hasDerivAt {f : ℝ → ℝ} (hf : Monotone f) :
    ∀ᵐ x, HasDerivAt f (rnDeriv hf.stieltjesFunction.measure volume x).toReal x 
goal: ∀ᵐ (x : ℝ), HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv MeasureTheory.volume x).toReal x

full_name: Monotone.ae_differentiableAt
code: /-- A monotone real function is differentiable Lebesgue-almost everywhere. -/
theorem Monotone.ae_differentiableAt {f : ℝ → ℝ} (hf : Monotone f) :
    ∀ᵐ x, DifferentiableAt ℝ f x 
goal: ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x

full_name: MonotoneOn.ae_differentiableWithinAt_of_mem
code: /-- A real function which is monotone on a set is differentiable Lebesgue-almost everywhere on
this set. This version does not assume that `s` is measurable. For a formulation with
`volume.restrict s` assuming that `s` is measurable, see `MonotoneOn.ae_differentiableWithinAt`.
-/
theorem MonotoneOn.ae_differentiableWithinAt_of_mem {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s) :
    ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x

full_name: MonotoneOn.ae_differentiableWithinAt
code: /-- A real function which is monotone on a set is differentiable Lebesgue-almost everywhere on
this set. This version assumes that `s` is measurable and uses `volume.restrict s`.
For a formulation without measurability assumption,
see `MonotoneOn.ae_differentiableWithinAt_of_mem`. -/
theorem MonotoneOn.ae_differentiableWithinAt {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s)
    (hs : MeasurableSet s) : ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x 
goal: ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x