File: Mathlib/Analysis/Calculus/ContDiff/RCLike.lean

full_name: HasFTaylorSeriesUpToOn.hasStrictFDerivAt
code: /-- If a function has a Taylor series at order at least 1, then at points in the interior of the
    domain of definition, the term of order 1 of this series is a strict derivative of `f`. -/
theorem HasFTaylorSeriesUpToOn.hasStrictFDerivAt {s : Set E'} {f : E' → F'} {x : E'}
    {p : E' → FormalMultilinearSeries 𝕂 E' F'} (hf : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
    (hs : s ∈ 𝓝 x) : HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕂 E' F') (p x 1)) x 
goal: HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕂 E' F') (p x 1)) x

full_name: ContDiffAt.hasStrictFDerivAt'
code: /-- If a function is `C^n` with `1 ≤ n` around a point, and its derivative at that point is given to
us as `f'`, then `f'` is also a strict derivative. -/
theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
    (hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
    HasStrictFDerivAt f f' x 
goal: HasStrictFDerivAt f f' x

full_name: ContDiffAt.hasStrictDerivAt'
code: /-- If a function is `C^n` with `1 ≤ n` around a point, and its derivative at that point is given to
us as `f'`, then `f'` is also a strict derivative. -/
theorem ContDiffAt.hasStrictDerivAt' {f : 𝕂 → F'} {f' : F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x)
    (hf' : HasDerivAt f f' x) (hn : 1 ≤ n) : HasStrictDerivAt f f' x 
goal: HasStrictDerivAt f f' x

full_name: ContDiffAt.hasStrictFDerivAt
code: /-- If a function is `C^n` with `1 ≤ n` around a point, then the derivative of `f` at this point
is also a strict derivative. -/
theorem ContDiffAt.hasStrictFDerivAt {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) :
    HasStrictFDerivAt f (fderiv 𝕂 f x) x 
goal: HasStrictFDerivAt f (fderiv 𝕂 f x) x

full_name: ContDiffAt.hasStrictDerivAt
code: /-- If a function is `C^n` with `1 ≤ n` around a point, then the derivative of `f` at this point
is also a strict derivative. -/
theorem ContDiffAt.hasStrictDerivAt {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) :
    HasStrictDerivAt f (deriv f x) x 
goal: HasStrictDerivAt f (deriv f x) x

full_name: ContDiff.hasStrictFDerivAt
code: /-- If a function is `C^n` with `1 ≤ n`, then the derivative of `f` is also a strict derivative. -/
theorem ContDiff.hasStrictFDerivAt {f : E' → F'} {x : E'} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
    HasStrictFDerivAt f (fderiv 𝕂 f x) x 
goal: HasStrictFDerivAt f (fderiv 𝕂 f x) x

full_name: ContDiff.hasStrictDerivAt
code: /-- If a function is `C^n` with `1 ≤ n`, then the derivative of `f` is also a strict derivative. -/
theorem ContDiff.hasStrictDerivAt {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
    HasStrictDerivAt f (deriv f x) x 
goal: HasStrictDerivAt f (deriv f x) x

full_name: HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt
code: /-- If `f` has a formal Taylor series `p` up to order `1` on `{x} ∪ s`, where `s` is a convex set,
and `‖p x 1‖₊ < K`, then `f` is `K`-Lipschitz in a neighborhood of `x` within `s`. -/
theorem HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt {E F : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
    {p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E}
    (hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) (K : ℝ≥0)
    (hK : ‖p x 1‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
goal: ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

full_name: HasFTaylorSeriesUpToOn.exists_lipschitzOnWith
code: /-- If `f` has a formal Taylor series `p` up to order `1` on `{x} ∪ s`, where `s` is a convex set,
then `f` is Lipschitz in a neighborhood of `x` within `s`. -/
theorem HasFTaylorSeriesUpToOn.exists_lipschitzOnWith {E F : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
    {p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E}
    (hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) :
    ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
goal: ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

full_name: ContDiffWithinAt.exists_lipschitzOnWith
code: /-- If `f` is `C^1` within a convex set `s` at `x`, then it is Lipschitz on a neighborhood of `x`
within `s`. -/
theorem ContDiffWithinAt.exists_lipschitzOnWith {E F : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {s : Set E} {x : E}
    (hf : ContDiffWithinAt ℝ 1 f s x) (hs : Convex ℝ s) :
    ∃ K : ℝ≥0, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
goal: ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

full_name: ContDiffAt.exists_lipschitzOnWith_of_nnnorm_lt
code: /-- If `f` is `C^1` at `x` and `K > ‖fderiv 𝕂 f x‖`, then `f` is `K`-Lipschitz in a neighborhood of
`x`. -/
theorem ContDiffAt.exists_lipschitzOnWith_of_nnnorm_lt {f : E' → F'} {x : E'}
    (hf : ContDiffAt 𝕂 1 f x) (K : ℝ≥0) (hK : ‖fderiv 𝕂 f x‖₊ < K) :
    ∃ t ∈ 𝓝 x, LipschitzOnWith K f t 
goal: ∃ t ∈ 𝓝 x, LipschitzOnWith K f t

full_name: ContDiffAt.exists_lipschitzOnWith
code: /-- If `f` is `C^1` at `x`, then `f` is Lipschitz in a neighborhood of `x`. -/
theorem ContDiffAt.exists_lipschitzOnWith {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 1 f x) :
    ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t 
goal: ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t

full_name: ContDiff.locallyLipschitz
code: /-- If `f` is `C^1`, it is locally Lipschitz. -/
lemma ContDiff.locallyLipschitz {f : E' → F'} (hf : ContDiff 𝕂 1 f) : LocallyLipschitz f 
goal: LocallyLipschitz f

full_name: ContDiff.lipschitzWith_of_hasCompactSupport
code: /-- A `C^1` function with compact support is Lipschitz. -/
theorem ContDiff.lipschitzWith_of_hasCompactSupport {f : E' → F'} {n : ℕ∞}
    (hf : HasCompactSupport f) (h'f : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
    ∃ C, LipschitzWith C f 
goal: ∃ C, LipschitzWith C f