Composite of Continuous Mappings is Continuous/Point

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Theorem

Let $T_1, T_2, T_3$ be topological spaces.

Let the function $f : T_1 \to T_2$ be continuous at $x$.

Let the function $g : T_2 \to T_3$ be continuous at $f\left({x}\right)$.

Then the function $g \circ f : T_1 \to T_3$ is continuous at $x$.


Proof

Let $N$ be any neighborhood of $\left({g \circ f}\right) \left({x}\right)$.

By the definition of continuity at a point:

there exists a neighborhood $L$ of $f \left({x}\right)$ such that $g \left({L}\right) \subseteq N$

and

there exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq L$.

Thus $\left({g \circ f}\right) \left({M}\right) \subseteq g \left({L}\right) \subseteq N$, as desired.

$\blacksquare$