English
A bi-parameter homomorphism with continuous at 1 in both variables and a bilinear-like structure is continuous on the product group.
Русский
Двойной гомоморфизм, непрерывный в одной точке, несовместим с непрерывностью на произведении, порождает непрерывность во всём произведении.
LaTeX
$$$\text{continuous_of_continuousAt_one₂}$$$
Lean4
/-- A monoid homomorphism (a bundled morphism of a type that implements `MonoidHomClass`) from a
topological group to a topological monoid is continuous provided that it is continuous at one. See
also `uniformContinuous_of_continuousAt_one`. -/
@[to_additive /-- An additive monoid homomorphism (a bundled morphism of a type that implements
`AddMonoidHomClass`) from an additive topological group to an additive topological monoid is
continuous provided that it is continuous at zero. See also
`uniformContinuous_of_continuousAt_zero`. -/
]
theorem continuous_of_continuousAt_one {M hom : Type*} [MulOneClass M] [TopologicalSpace M] [ContinuousMul M]
[FunLike hom G M] [MonoidHomClass hom G M] (f : hom) (hf : ContinuousAt f 1) : Continuous f :=
continuous_iff_continuousAt.2 fun x => by
simpa only [ContinuousAt, ← map_mul_left_nhds_one x, tendsto_map'_iff, Function.comp_def, map_mul, map_one,
mul_one] using hf.tendsto.const_mul (f x)
