Definition:Discontinuous Mapping/Topological Space/Point

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Definition

Let $T_1 = \struct {A_1, \tau_1}$ and $T_2 = \struct {A_2, \tau_2}$ be topological spaces.

Let $f: A_1 \to A_2$ $x \in T_1$ be a mapping from $A_1$ to $A_2$.


Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $\map f x$ there exists a neighborhood $M$ of $x$ such that $f \sqbrk M \subseteq N$.

Therefore, $f$ is discontinuous at $x$ if for some neighborhood $N$ of $\map f x$ and every neighborhood $M$ of $x$:

$f \sqbrk M \nsubseteq N$

The point $x$ is called a discontinuity of $f$.


Also see


Sources