Continuous Function

Context

The definitions of continuity in terms of metric spaces

Statements

Definition (@Men90, Definition 3.2). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A mapping $f:X\to Y$ is said to be continuous at $a\in X$ if, for every $\epsilon >0$, there exists $\delta >0$, such that the inequality $d_Y(f(x),f(a))<\epsilon$ implies $d_X(x,a)<\delta$.

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and topological spaces are equivalent

Statements

Definition (@Tu11 pp. 327, @Bou95, Chapter 2, Definition 1). Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces. A mapping $f:X\to Y$ is said to be continuous at a point $x\in X$ if, for every topological neighborhood $N_Y$ of $f(x)\subset Y$, there exists a topological neighborhood $N_X$ of $x\in X$ such that the relation $x\in N_X$ implies $f(x)\in N_Y$. In addition, $f$ is continuous on $X$ if it is continuous at every point of $X$.

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as the following theorem shows.

Statements

Theorem (@Men90, Theorem 6.3). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, the function $f:(X,d_X)\to (Y,d_Y)$ is a metrical continuous function if, and only, if it is a topological continuous function.

Remarks

Another definition for a continuous function can be done in terms of sequences as follows

Statements

Theorem (@Men90, Theorem 5.4). Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A function $f:X\to Y$ is continuous at a point $x\in X$ if and only if, for a convergent sequence $\{x_i{\}}_{i\in \mathbb{N}}\subset X$, the sequence $\{f(x_i){\}}_{i\in \mathbb{N}}$ is also convergent.

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