piecewise — Mathlib

English

If h,f,g satisfy the hypotheses of continuous-on across the frontier with equalities on the boundary, then the piecewise function t.f g is continuous on t.ite s s'.

Русский

Если функции f,g удовлетворяют гипотезам непрерывности на границе и совпадают на границе, то t.piecewise f g непрерывна на t.ite s s'.

LaTeX

$$$\text{ContinuousOn piecewise}'(f,g,t,s,s')$ under the given boundary equalities.$$

Lean4

theorem piecewise [∀ a, Decidable (a ∈ s)] (hs : ∀ a ∈ frontier s, f a = g a) (hf : Continuous f) (hg : Continuous g) :
    Continuous (piecewise s f g) :=
  hf.if hs hg

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