specialization_monotone — Mathlib

English

A continuous function is monotone with respect to the specialization preorders on domain and codomain.

Русский

Непрерывная функция монотонна относительно порядков специализации на области и кодomaine.

LaTeX

$$$$ \text{If } f\text{ is continuous, then } f \text{ is monotone: } x \rightsquigarrow y \Rightarrow f(x) \rightsquigarrow f(y). $$$$

Lean4

/-- A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. -/
theorem specialization_monotone (hf : Continuous f) : Monotone f := fun _ _ h => h.map hf

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