norm' — Mathlib

English

If f is continuous within a set s, then the norm of f is continuous within s.

Русский

Если f непрерывна на множестве s, то ‖f‖ непрерывна на s.

LaTeX

$$$ \text{ContinuousWithinAt}(f,s) \Rightarrow \text{ContinuousWithinAt}(\lambda x. \|f x\|, s). $$$

Lean4

@[to_additive ContinuousWithinAt.norm]
theorem norm' {s : Set α} {a : α} (h : ContinuousWithinAt f s a) : ContinuousWithinAt (fun x => ‖f x‖) s a :=
  Tendsto.norm' h

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