Continuity

Definition: Continuity at a Point

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

A function $f:X\to Y$ is continuous at $x\in X$ iff for each neighbourhood $V$ of $f(x)$ there exists a neighbourhood $U$ of $x$ such that $f(U)\subset V$.

Definition: Continuity

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

A function $f:X\to Y$ is continuous on $X$ or simply continuous iff it is continuous at each $x\in X$.

Continuity Criteria

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 $X$.

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 $X$.

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 $x\in X$ has a neighbourhood $N$ such that the restriction $f{|}_N$ is continuous.

PROOF

TODO

Properties

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 $(X,{\tau }_X)$, $(Y,{\tau }_Y)$ and $(Z,{\tau }_Z)$ 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

We need to prove only that if $U$ is open in $(Z,{\tau }_Z)$, then its inverse image $(g\circ f{)}^{-1}(U)$ is open in $(X,{\tau }_X)$.

Let $U$ be open in $(Z,{\tau }_Z)$. We know that $(g\circ f{)}^{-1}(U)=f^{-1}(g^{-1}(U))$. Since $g$ is and $U$ is open in $(Z,{\tau }_Z)$, the aforementioned theorem tells us that $g^{-1}(U)$ is open in $(Y,{\tau }_Y)$. Analogously, since $f$ is and $g^{-1}(U)$ is open in $(Y,{\tau }_Y)$, the theorem tells us that $f^{-1}(g^{-1}(U))$ is in $(X,{\tau }_X)$.

Theorem: Continuity of Restrictions

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

If $f:X\to Y$ is continuous and $S$ is an open subset of $X$, then the restriction $f{|}_S$ is also continuous.

PROOF

TODO

Intermediate Value Theorem

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

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

PROOF

TODO