feat(Convex/Gauge): add continuousAt_gauge (#9911)

Mathlib/Analysis/Convex/Gauge.lean

@@ -405,10 +405,9 @@ variable [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E]
 /-- If `s` is a convex neighborhood of the origin in a topological real vector space, then `gauge s`
 is continuous. If the ambient space is a normed space, then `gauge s` is Lipschitz continuous, see
 `Convex.lipschitz_gauge`. -/
-theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous (gauge s) := by
+theorem continuousAt_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : ContinuousAt (gauge s) x := by
   have ha : Absorbent ℝ s := absorbent_nhds_zero hs₀
-  simp only [continuous_iff_continuousAt, ContinuousAt, (nhds_basis_Icc_pos _).tendsto_right_iff]
-  intro x ε hε₀
+  refine (nhds_basis_Icc_pos _).tendsto_right_iff.2 fun ε hε₀ ↦ ?_
   rw [← map_add_left_nhds_zero, eventually_map]
   have : ε • s ∩ -(ε • s) ∈ 𝓝 0
   · exact inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
@@ -424,6 +423,13 @@ theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous
       gauge s (x + y) ≤ gauge s x + gauge s y := gauge_add_le hc ha _ _
       _ ≤ gauge s x + ε := add_le_add_left (gauge_le_of_mem hε₀.le hy.1) _
 
+/-- If `s` is a convex neighborhood of the origin in a topological real vector space, then `gauge s`
+is continuous. If the ambient space is a normed space, then `gauge s` is Lipschitz continuous, see
+`Convex.lipschitz_gauge`. -/
+@[continuity]
+theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous (gauge s) :=
+  continuous_iff_continuousAt.2 fun _ ↦ continuousAt_gauge hc hs₀
+
 theorem gauge_lt_one_eq_interior (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
     { x | gauge s x < 1 } = interior s := by
   refine Subset.antisymm (fun x hx ↦ ?_) (interior_subset_gauge_lt_one s)