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feat(Topology/Continuous): Continuous.preimage_mem_nhds #38011

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feat(Topology/Continuous): Continuous.preimage_mem_nhds #38011

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4 changes: 4 additions & 0 deletions Mathlib/Topology/Continuous.lean
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Expand Up @@ -149,6 +149,10 @@ theorem Continuous.tendsto' (hf : Continuous f) (x : X) (y : Y) (h : f x = y) :
theorem Continuous.continuousAt (h : Continuous f) : ContinuousAt f x :=
h.tendsto x

theorem Continuous.preimage_mem_nhds {t : Set Y} (h : Continuous f)
(ht : t ∈ 𝓝 (f x)) : f ⁻¹' t ∈ 𝓝 x :=
h.continuousAt.preimage_mem_nhds ht

theorem continuous_iff_continuousAt : Continuous f ↔ ∀ x, ContinuousAt f x :=
⟨Continuous.tendsto, fun hf => continuous_def.2 fun _U hU => isOpen_iff_mem_nhds.2 fun x hx =>
hf x <| hU.mem_nhds hx⟩
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