mem_closure_image — Mathlib

English

If f is continuous within s at x and (g x, h x) equals y, then f(g x, h x) is in the closure of the image; more precisely, composing with a map of the limit point preserves closure.

Русский

Если f непрерывна внутри s в x и пары (g x, h x) сходится к y, то f(g x, h x) лежит в замыкании образа.

LaTeX

$$$\text{ContinuousAt } f (g x, h x) \to \text{ContinuousWithinAt } g s x \to \text{ContinuousWithinAt } h s x \to \text{ContinuousWithinAt } f (g x, h x)$$$

Lean4

theorem mem_closure_image (h : ContinuousWithinAt f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) :=
  haveI := mem_closure_iff_nhdsWithin_neBot.1 hx
  mem_closure_of_tendsto h <| mem_of_superset self_mem_nhdsWithin (subset_preimage_image f s)

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