feat: Add a few continuity lemmas for products (#10820)

1. `ContinuousAt.comp₂`, `ContinuousAt.comp₂_continuousWithinAt`, and `_of_eq` versions.
2. `ContinuousAt.along_{fst,snd}`: Continuous functions are continuous in their first and second arguments.

Mathlib/Analysis/Analytic/Constructions.lean

@@ -119,24 +119,28 @@ theorem AnalyticOn.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {s : S
   fun _ xt ↦ (ha _ (m _ xt)).comp₂ (fa _ xt) (ga _ xt)
 
 /-- Analytic functions on products are analytic in the first coordinate -/
-theorem AnalyticAt.along_fst {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
+theorem AnalyticAt.curry_left {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
     AnalyticAt 𝕜 (fun x ↦ f (x, p.2)) p.1 :=
   AnalyticAt.comp₂ fa (analyticAt_id _ _) analyticAt_const
+alias AnalyticAt.along_fst := AnalyticAt.curry_left
 
 /-- Analytic functions on products are analytic in the second coordinate -/
-theorem AnalyticAt.along_snd {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
+theorem AnalyticAt.curry_right {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
     AnalyticAt 𝕜 (fun y ↦ f (p.1, y)) p.2 :=
   AnalyticAt.comp₂ fa analyticAt_const (analyticAt_id _ _)
+alias AnalyticAt.along_snd := AnalyticAt.curry_right
 
 /-- Analytic functions on products are analytic in the first coordinate -/
-theorem AnalyticOn.along_fst {f : E × F → G} {s : Set (E × F)} {y : F} (fa : AnalyticOn 𝕜 f s) :
+theorem AnalyticOn.curry_left {f : E × F → G} {s : Set (E × F)} {y : F} (fa : AnalyticOn 𝕜 f s) :
     AnalyticOn 𝕜 (fun x ↦ f (x, y)) {x | (x, y) ∈ s} :=
   fun x m ↦ (fa (x, y) m).along_fst
+alias AnalyticOn.along_fst := AnalyticOn.curry_left
 
 /-- Analytic functions on products are analytic in the second coordinate -/
-theorem AnalyticOn.along_snd {f : E × F → G} {x : E} {s : Set (E × F)} (fa : AnalyticOn 𝕜 f s) :
+theorem AnalyticOn.curry_right {f : E × F → G} {x : E} {s : Set (E × F)} (fa : AnalyticOn 𝕜 f s) :
     AnalyticOn 𝕜 (fun y ↦ f (x, y)) {y | (x, y) ∈ s} :=
   fun y m ↦ (fa (x, y) m).along_snd
+alias AnalyticOn.along_snd := AnalyticOn.curry_right
 
 /-!
 ### Arithmetic on analytic functions

Mathlib/Analysis/Calculus/FDeriv/Measurable.lean

@@ -853,7 +853,7 @@ lemma isOpen_A_with_param {r s : ℝ} (hf : Continuous f.uncurry) (L : E →L[
   simp only [A, half_lt_self_iff, not_lt, mem_Ioc, mem_ball, map_sub, mem_setOf_eq]
   apply isOpen_iff_mem_nhds.2
   rintro ⟨a, x⟩ ⟨r', ⟨Irr', Ir'r⟩, hr⟩
-  have ha : Continuous (f a) := continuous_uncurry_left a hf
+  have ha : Continuous (f a) := hf.uncurry_left a
   rcases exists_between Irr' with ⟨t, hrt, htr'⟩
   rcases exists_between hrt with ⟨t', hrt', ht't⟩
   obtain ⟨b, b_lt, hb⟩ : ∃ b, b < s * r ∧ ∀ y ∈ closedBall x t, ∀ z ∈ closedBall x t,

Mathlib/Topology/Constructions.lean

@@ -508,18 +508,22 @@ lemma isClosedMap_swap : IsClosedMap (Prod.swap : X × Y → Y × X) := fun s hs
   rw [image_swap_eq_preimage_swap]
   exact hs.preimage continuous_swap
 
-theorem continuous_uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) :
+theorem Continuous.uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) :
     Continuous (f x) :=
   h.comp (Continuous.Prod.mk _)
-#align continuous_uncurry_left continuous_uncurry_left
+#align continuous_uncurry_left Continuous.uncurry_left
 
-theorem continuous_uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) :
+theorem Continuous.uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) :
     Continuous fun a => f a y :=
   h.comp (Continuous.Prod.mk_left _)
-#align continuous_uncurry_right continuous_uncurry_right
+#align continuous_uncurry_right Continuous.uncurry_right
+
+-- 2024-03-09
+@[deprecated] alias continuous_uncurry_left := Continuous.uncurry_left
+@[deprecated] alias continuous_uncurry_right := Continuous.uncurry_right
 
 theorem continuous_curry {g : X × Y → Z} (x : X) (h : Continuous g) : Continuous (curry g x) :=
-  continuous_uncurry_left x h
+  Continuous.uncurry_left x h
 #align continuous_curry continuous_curry
 
 theorem IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
@@ -636,6 +640,29 @@ theorem ContinuousAt.prod_map' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf :
   hf.fst'.prod hg.snd'
 #align continuous_at.prod_map' ContinuousAt.prod_map'
 
+theorem ContinuousAt.comp₂ {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X}
+    (hf : ContinuousAt f (g x, h x)) (hg : ContinuousAt g x) (hh : ContinuousAt h x) :
+    ContinuousAt (fun x ↦ f (g x, h x)) x :=
+  ContinuousAt.comp hf (hg.prod hh)
+
+theorem ContinuousAt.comp₂_of_eq {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} {y : Y × Z}
+    (hf : ContinuousAt f y) (hg : ContinuousAt g x) (hh : ContinuousAt h x) (e : (g x, h x) = y) :
+    ContinuousAt (fun x ↦ f (g x, h x)) x := by
+  rw [← e] at hf
+  exact hf.comp₂ hg hh
+
+/-- Continuous functions on products are continuous in their first argument -/
+theorem Continuous.curry_left {f : X × Y → Z} (hf : Continuous f) {y : Y} :
+    Continuous fun x ↦ f (x, y) :=
+  hf.comp (continuous_id.prod_mk continuous_const)
+alias Continuous.along_fst := Continuous.curry_left
+
+/-- Continuous functions on products are continuous in their second argument -/
+theorem Continuous.curry_right {f : X × Y → Z} (hf : Continuous f) {x : X} :
+    Continuous fun y ↦ f (x, y) :=
+  hf.comp (continuous_const.prod_mk continuous_id)
+alias Continuous.along_snd := Continuous.curry_right
+
 -- todo: reformulate using `Set.image2`
 -- todo: prove a version of `generateFrom_union` with `image2 (∩) s t` in the LHS and use it here
 theorem prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : Set (Set Y)}

Mathlib/Topology/ContinuousOn.lean

@@ -1152,6 +1152,19 @@ theorem ContinuousOn.prod {f : α → β} {g : α → γ} {s : Set α} (hf : Con
   ContinuousWithinAt.prod (hf x hx) (hg x hx)
 #align continuous_on.prod ContinuousOn.prod
 
+theorem ContinuousAt.comp₂_continuousWithinAt {f : β × γ → δ} {g : α → β} {h : α → γ} {x : α}
+    {s : Set α} (hf : ContinuousAt f (g x, h x)) (hg : ContinuousWithinAt g s x)
+    (hh : ContinuousWithinAt h s x) :
+    ContinuousWithinAt (fun x ↦ f (g x, h x)) s x :=
+  ContinuousAt.comp_continuousWithinAt hf (hg.prod hh)
+
+theorem ContinuousAt.comp₂_continuousWithinAt_of_eq {f : β × γ → δ} {g : α → β}
+    {h : α → γ} {x : α} {s : Set α} {y : β × γ} (hf : ContinuousAt f y)
+    (hg : ContinuousWithinAt g s x) (hh : ContinuousWithinAt h s x) (e : (g x, h x) = y) :
+    ContinuousWithinAt (fun x ↦ f (g x, h x)) s x := by
+  rw [← e] at hf
+  exact hf.comp₂_continuousWithinAt hg hh
+
 theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
     {x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by
   simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]; rfl

Mathlib/Topology/Separation.lean

@@ -1605,8 +1605,7 @@ theorem eqOn_closure₂' [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y →
 theorem eqOn_closure₂ [T2Space Z] {s : Set X} {t : Set Y} {f g : X → Y → Z}
     (h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y) (hf : Continuous (uncurry f))
     (hg : Continuous (uncurry g)) : ∀ x ∈ closure s, ∀ y ∈ closure t, f x y = g x y :=
-  eqOn_closure₂' h (fun x => continuous_uncurry_left x hf) (fun x => continuous_uncurry_right x hf)
-    (fun y => continuous_uncurry_left y hg) fun y => continuous_uncurry_right y hg
+  eqOn_closure₂' h hf.uncurry_left hf.uncurry_right hg.uncurry_left hg.uncurry_right
 #align eq_on_closure₂ eqOn_closure₂
 
 /-- If `f x = g x` for all `x ∈ s` and `f`, `g` are continuous on `t`, `s ⊆ t ⊆ closure s`, then