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Continuous

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Definition
RealAnal

Let (X,d),(Y,ρ)(X,d),(Y,\rho) be metric spaces and let f:X→Yf:X\to Y. We say that ff is continuous at x∈Xx\in X if and only if for every sequence (xn)n∈N(x_{n})_{n\in\mathbb{N}} s.t. xn→xx_{n}\to x. We have f(xn)→f(x)f(x_{n})\to f(x)The function is continuous if it is continuous ∀x∈X\forall x\in X.

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}

Definition (Calculus)

The function ff is continuous at some point cc of its domain if the limit of f(x)f(x) as xx approaches cc through the domain of ff, exists and is equal to f(c)f(c). In mathematical notation, this is written as lim⁡x→cf(x)=f(c)\lim_{ x \to c } f(x)=f(c)