Continuity

Definition: Continuity

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

A function $f:X\to Y$ is continuous at $p\in X$ if for each neighborhood $V$ of $f(p)$ there exists a neighborhood $U$ of $p$ such that

$$x\in U⟹f(x)\in V$$

If $f$ is continuous at every $p\in S\subseteq X$, then we say that $f$ is continuous on $S$ or simply continuous if $S=X$.

Theorem: Continuity via Openness

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

A function $f:X\to Y$ is continuous if and only if the inverse image of each open subset of $Y$ is an open subset of $X$.

$$U\in {\tau }_Y⟹f^{-1}(U)\in {\tau }_X$$

PROOF

TODO

Theorem: Continuity via Closedness

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

A function $f:X\to Y$ is continuous if and only if the inverse image of each closed subset of $Y$ is a closed subset of $X$.

$$Y\setminus C\in {\tau }_Y⟹X\setminus f^{-1}(C)\in {\tau }_X$$

PROOF

TODO

Theorem: Local Criterion

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

A function $f:X\to Y$ is continuous if and only if each point $p\in X$ has a neighborhood $N(p)$ on which $f$ is continuous.

PROOF

TODO

Extreme Value Theorem

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

If $(X,{\tau }_X)$ is compact, then its image $f(X)$ under every continuous function $f:X\to Y$ is also compact.

PROOF

TODO

Theorem: Continuity of Composition

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

If $f:X\to Y$ and $g:Y\to Z$ are continuous, then so is their composition $g\circ f:X\to Z$.

PROOF

TODO

Intermediate Value Theorem

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be topological spaces.

If $(X,{\tau }_X)$ is connected, then its image $f(X)$ under every continuous function $f:X\to Y$ is also connected.

PROOF

TODO