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Continuous

Definition (Continuous)

Let (X,TX)(X,\mathscr{T}_{X}) and (Y,TY)(Y,\mathscr{T}_{Y}) be topological spaces, and let f:X→Yf:X\to Y be a map.

  1. The map ff is continuous at x0x_{0} for x0∈Xx_{0}\in X if, for each neighbourhood V\mathcal{V} of f(x0)f(x_{0}), there exists a neighbourhood U\mathcal{U} of x0x_{0} such that f(U)⊂V.f(\mathcal{U})\subset \mathcal{V}.
  2. If ff is continuous at each x∈Xx\in X then it is continuous. One may verify that ff is continuous if and only if ∀O∈TY:f−1(O)∈TX\forall \mathcal{O}\in \mathscr{T}_{Y}: f^{-1}(\mathcal{O})\in \mathscr{T}_{X}

Theorem (Equivalent definitions)

The following are equivalent for a function ff from a Metric Space (X,d)(X,d) to a Metric Space (Y,ρ)(Y,\rho):

  1. ff is continuous.
  2. ∀x∈X,∀ϵ>0,∃δ>0:∀y∈X,d(x,y)<δ  ⟹  ρ(f(x),f(y))<ϵ\forall x\in X,\forall\epsilon>0,\exists\delta>0:\forall y\in X,d(x,y)<\delta\implies\rho(f(x),f(y))<\epsilon
  3. Let (X,T),(Y,TY)(X,\mathscr{T}),(Y,\mathscr{T}_{Y}) be the metric topologies defined by their respective metric spaces then ∀A∈TY,f−1(A)∈T\forall A\in\mathscr{T}_{Y},f^{-1}(A)\in\mathscr{T}

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