Equivalence of Definitions of Continuous Mapping between Topological Spaces/Point

Theorem
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.

Let $x \in S_1$.

$(2)$ implies $(3)$
Let $f$ be a continuous mapping defined using neighborhoods.

Then by definition:
 * For every neighborhood $N$ of $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.

Let $\FF$ be a filter on $T_1$ that converges to $x$.

Let $N \subseteq S_2$ be a neighborhood of $\map f x$ in $T_2$.

Then $f^{-1} \sqbrk N$ is a neighborhood of $x$ in $T_1$.

Because $\FF$ converges to $x$, this implies that $f^{-1} \sqbrk N \in \FF$.

By the definition of $f \sqbrk \FF$ it follows that $N \in f \sqbrk \FF$.

Thus $f \sqbrk \FF$ converges to $\map f x$.

That is, $f$ is a continuous mapping defined using filters.

$(3)$ implies $(2)$
Let $f$ be a continuous mapping defined using filters.

That is:
 * for any filter $\FF$ on $T_1$ that converges to $x$, the corresponding image filter $f \sqbrk \FF$ converges to $\map f x$.

Let $\UU_x$ be the set defined as:
 * $\UU_x := \leftset {M \subseteq S_1: M}$ is a neighborhood of $\rightset x$

We have that $\UU_x$ is a filter on $S_1$.

Let $\FF := \UU_x$.

By the definition of convergent filter, $\FF$ converges to $x$.

By assumption this implies that $f \sqbrk \FF$ converges to $\map f x$.

Now let $N \subseteq S_2$ be a neighborhood of $\map f x$.

Then:
 * $N \in f \sqbrk \FF$

By definition of image of subset under mapping:
 * $f^{-1} \sqbrk N \in \FF = \UU_x$

Thus $f^{-1} \sqbrk N$ is a neighborhood of $x$.

Therefore $f$ is continuous at $x$.