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202209300929 Theorem 4.1.3

Prove Theorem 4.1.3.

Let (X,d) and (Y,\rho ) be metric spaces and f:X\to Y be a function. Then, f is continuous at a point x_0 if and only if f(x_n)\to f(x_0), for every sequence \{ x_n\} \subset X with x_n\to x_0.

Extracted from K. J. Pawan & A. Khalil. (2004). Metric Spaces.

Background.

Define, by open spheres, continuity of a function f at a point x_0:

\forall\, x,\exists\, x_0\in X, \forall\, \epsilon >0, \exists\,\delta >0, d(x,x_0)<\delta\Rightarrow \rho (f(x),f(x_0))<\epsilon

in other words,

x\in S_\delta (x_0)\Rightarrow f(x)\in S_\epsilon (f(x_0)),

or the same,

f(S_\delta (x_0))\subset S_\epsilon (f(x_0)).

Proof.

\begin{aligned} & f\textrm{ is continuous at }x_0 \\ \Leftrightarrow\enspace & f(S_\delta (x_0))\subset S_\epsilon (f(x_0))\textrm{ for some } \delta>0\textrm{ and any }\epsilon >0 \\ & \\ & \textrm{for every sequence }\{ x_n\}\subset X\textrm{ with }x_n\textrm{ tends to }x_0 \\ \Leftrightarrow\enspace & \textrm{there exists an }N>0\textrm{ such that }x_n\in S_{\delta}(x_0)\textrm{ for all }n>N \\ \end{aligned}

\begin{aligned} & x_n\in S_\delta (x_0) \Longrightarrow f(x_n)\in f(S_\delta (x_0))\subset S_\epsilon (f(x_0)) \\ \Leftrightarrow\enspace & f(x_n) \to f(x_0) \\ \end{aligned}