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Proposition: Equivalent Notions of Continuous Functions
Let $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ be topological spaces.
The following definitions of a continuous function $f:X\to Y$ are equivalent:
 The inverse image $f^{1}[B]$ of every open set $B$ in $Y$ is open in $X.$
 The inverse image $f^{1}[B]$ of every closed set $B$ in $Y$ is closed in $X.$
 The image of the closure of a subset $A\subset X$ is contained in the closure of the image of this subset, formally $f[A^]\subseteq f[A]^.$
 For each $x\in X$ and each neighborhood $N_{f(x)}$ of $f(x)$ there exists a neighborhood $N_x$ of $x$ such that $f[N_x]\subseteq N_{f(x)}.$
Notes
 If the 4th condition is fulfilled for a particular point $x,$ $f$ is said to be continuous at the point $x.$
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References
Bibliography
 Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
 Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition