File: Mathlib/Analysis/Oscillation.lean

full_name: oscillation
code: /-- The oscillation of `f : E → F` at `x`. -/
noncomputable def oscillation [TopologicalSpace E] (f : E → F) (x : E) : ENNReal 
goal: ℝ≥0∞

full_name: oscillationWithin
code: /-- The oscillation of `f : E → F` within `D` at `x`. -/
noncomputable def oscillationWithin [TopologicalSpace E] (f : E → F) (D : Set E) (x : E) :
  ENNReal 
goal: ℝ≥0∞

full_name: oscillationWithin_nhd_eq_oscillation
code: /-- The oscillation of `f` at `x` within a neighborhood `D` of `x` is equal to `oscillation f x` -/
theorem oscillationWithin_nhd_eq_oscillation [TopologicalSpace E] (f : E → F) (D : Set E) (x : E)
    (hD : D ∈ 𝓝 x) : oscillationWithin f D x = oscillation f x 
goal: oscillationWithin f D x = oscillation f x

full_name: oscillationWithin_univ_eq_oscillation
code: /-- The oscillation of `f` at `x` within `univ` is equal to `oscillation f x` -/
theorem oscillationWithin_univ_eq_oscillation [TopologicalSpace E] (f : E → F) (x : E) :
    oscillationWithin f univ x = oscillation f x 
goal: oscillationWithin f Set.univ x = oscillation f x

full_name: ContinuousWithinAt.oscillationWithin_eq_zero
code: theorem oscillationWithin_eq_zero [TopologicalSpace E] {f : E → F} {D : Set E}
    {x : E} (hf : ContinuousWithinAt f D x) : oscillationWithin f D x = 0 
goal: oscillationWithin f D x = 0

full_name: ContinuousAt.oscillation_eq_zero
code: theorem oscillation_eq_zero [TopologicalSpace E] {f : E → F} {x : E} (hf : ContinuousAt f x) :
    oscillation f x = 0 
goal: oscillation f x = 0

full_name: OscillationWithin.eq_zero_iff_continuousWithinAt
code: /-- The oscillation within `D` of `f` at `x ∈ D` is 0 if and only if `ContinuousWithinAt f D x`. -/
theorem eq_zero_iff_continuousWithinAt [TopologicalSpace E] (f : E → F) {D : Set E}
    {x : E} (xD : x ∈ D) : oscillationWithin f D x = 0 ↔ ContinuousWithinAt f D x 
goal: oscillationWithin f D x = 0 ↔ ContinuousWithinAt f D x

full_name: Oscillation.eq_zero_iff_continuousAt
code: /-- The oscillation of `f` at `x` is 0 if and only if `f` is continuous at `x`. -/
theorem eq_zero_iff_continuousAt [TopologicalSpace E] (f : E → F) (x : E) :
    oscillation f x = 0 ↔ ContinuousAt f x 
goal: oscillation f x = 0 ↔ ContinuousAt f x

full_name: IsCompact.uniform_oscillationWithin
code: /-- If `oscillationWithin f D x < ε` at every `x` in a compact set `K`, then there exists `δ > 0`
such that the oscillation of `f` on `ball x δ ∩ D` is less than `ε` for every `x` in `K`. -/
theorem uniform_oscillationWithin (hK : ∀ x ∈ K, oscillationWithin f D x < ε) :
    ∃ δ > 0, ∀ x ∈ K, diam (f '' (ball x (ENNReal.ofReal δ) ∩ D)) ≤ ε 
goal: ∃ δ > 0, ∀ x ∈ K, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal δ) ∩ D)) ≤ ε

full_name: IsCompact.uniform_oscillation
code: /-- If `oscillation f x < ε` at every `x` in a compact set `K`, then there exists `δ > 0` such
that the oscillation of `f` on `ball x δ` is less than `ε` for every `x` in `K`. -/
theorem uniform_oscillation [PseudoEMetricSpace E] {K : Set E} (comp : IsCompact K)
    {f : E → F} {ε : ENNReal} (hK : ∀ x ∈ K, oscillation f x < ε) :
    ∃ δ > 0, ∀ x ∈ K, diam (f '' (ball x (ENNReal.ofReal δ))) ≤ ε 
goal: ∃ δ > 0, ∀ x ∈ K, EMetric.diam (f '' EMetric.ball x (ENNReal.ofReal δ)) ≤ ε