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How does the neighborhood definition of continuity work in a topological space?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, the neighborhood definition of continuity provides a way to describe functions that vary without abrupt breaks, generalizing the concept of continuity from mathematical analysis.

A function $f$ mapping a topological space $X$ into a topological space $Y$ is defined as continuous at a point $p \in X$ if, for any neighborhood $V$ of $f (p)$ in $Y$ , there exists a neighborhood $U$ of $p$ in $X$ such that the image of $U$ under $f$ is contained within $V$ —expressed as $f (U) \subseteq V$ .1

This definition essentially ensures that if you take a region around the output $f (p)$ , you can always find a corresponding region around the input $p$ that maps entirely into that output region. This approach is equivalent to the more abstract definition of continuity, which states that a function is continuous if the preimage of every open set in $Y$ is an open set in $X$ .12