Continuous functions

Question

• Fact (4) – why restrict domain and codomain? Is this an accurate takeaway? How is it different from (6)?

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Overview §

• Continuity in topology can be checked in terms of open or closed sets
• Subspace and product topology arise naturally from the definition of continuous function in topological spaces
• Continuous functions preserve topological properties

Informally, a function is uniformly continuous if it is continuous “in the same way” at every point—no matter how close the outputs are chosen to be, as long as the inputs are some close enough, then the outputs will be at least that close.

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In Euclidean space §

Definition: Continuous function in Euclidean space

A function $f:D\to \mathbb{R}$ is continuous if for all $x\in D$, $f$ is continuous at $x_0$.

$${\forall }_{x_0\in D}{\forall }_{ϵ>0}{\exists }_{\delta (x_0,ϵ)>0}{\forall }_{x\in D}[|x-x_0|<\delta ⟹|f(x)-f(x_0)|<ϵ]$$

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In metric spaces §

Definition: Continuous functions in metric spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. We say a function $f:X\to Y$ is:

• Continuous at $x_0\in X$ if for every $ϵ>0$, there exists $\delta >0$ such that

$$d_X(x,x_0)<\delta ⟹d_Y(f(x),f(x_0))<ϵ,$$

and $f$ continuous if this holds for all $x_0\in X$.

• Uniformly continuous if for all $ϵ>0$, there exists $\delta >0$ such that for any choice of $x,x_0\in X$, we have

$$d_X(x,x_0)<\delta ⟹d_Y(f(x),f(x_0))<ϵ.$$

Theorem: Continuity is equals uniform continuity on compact spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. If $f:X\to Y$ is continuous and $X$ is compact, then $f$ is uniformly continuous.

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In general topological spaces §

Definition: Continuous function in topological space

Let $X,Y$ be topological spaces. A function $f:X\to Y$ is continuous if for every open subset $V\subset Y$, the preimage set $f^{-1}(V)$ is an open subset of $X$.

Note that continuity can also be checked in terms of closed sets!

Theorem (Munkres 18.1): Equivalent statements of continuity

Let $f:X\to Y$ be a function between topological spaces $X,Y$. Then the following are equivalent:

• (a) $f$ is continuous.
• (b) For every closed set $B\subset Y$, the set $f^{-1}(B)$ is closed in $X$.
• (c) For all $x\in X$ and every neighborhood $V$ of (i.e., open set $V$ containing) $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U)\subseteq V$.
• (d) For every subset $A\subset X$, we have $f(\overline{A})\subset \overline{f(A)}$, where the overline denotes the closure of a set.

Note that condition (c) is similar to $ϵ-\delta$—why?

#wip Also note that it suffices to prove every basis and subbasis is open

Facts about continuous functions in topological spaces §

#wip compare to Munkres 18.2

Every constant function is continuous §

The subspace topology is the coarsest topology that makes the inclusion function continuous §

See also: Universal properties of topologies

Restrictions of the domain and restrictions or expansions of the range are continuous §

#concept-question What is the upshot?

The projection functions are continuous, and functions into product spaces are continuous if their coordinate functions are continuous §

Theorem: A function is continuous if and only if its coordinate functions are continuous

Let $X\times Y$ be a product space, and ${\pi }_1:X\times Y\to X$ defined by ${\pi }_1(x,y)=x$ and ${\pi }_2:X\times Y\to Y$ defined by ${\pi }_2(x,y)=y$ be the projection maps. A function into the product space $f:Z\to X\times Y$ is uniquely determined by the coordinate functions

$$f_1={\pi }_1\circ f\text{and}{f}_2={\pi }_2\circ f,$$

which satisfy $f(z)=(f_1(z),f_2(z))$. Then $f$ is continuous if and only if $f_1,f_2$ are continuous.

A function is continuous if for every open set that covers the space, the restriction to that open set is continuous §

#concept-question How is this different from 4?

The pasting lemma is a special case of this “local formulation of continuity.”

Lemma (Munkres 18.3): Pasting

Let $X=A\cup B$ for some $A,B\subseteq X$ that are both closed (or both open) in $X$, and let $f:A\to Y,g:B\to Y$ be continuous functions such that for all $x\in A\cap B$, we have $f(x)=g(x)$. Then $h:X\to Y$ defined by

$$h(x)=\{\begin{array}{r}f(x)\text{if }x\in A\\ g(x)\text{if }x\in B\end{array}$$

is continuous as well.

Continuous functions preserve limits §

Lemma ( Topology HW 3.6): Continuous functions preserve limits

If a sequence $\{x_n{\}}_{n\ge 1}$ in a topological space $X$ converges to some limit point $x\in X$, and $f:X\to Y$ is continuous, then $\{f(x_n){\}}_{n\ge 1}$ converges to $f(x)\in Y$.

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In general metrizable spaces §

Recall that a general topological space $(X,\mathcal{T})$ is metrizable if there exists some metric on $X$ which induces the topology $\mathcal{T}$. Specifically, the set of all open balls given by this metric is a basis for $\mathcal{T}$ on $X$.

Lemma (Munkres 21.1): \epsilon-\delta definition of continuity in metrizable spaces

Let $(X,d_X),(Y,d_Y)$ be metric spaces. Then a function $f:X\to Y$ is continuous if and only for all $x\in X$ and all $ϵ>0$, there exists $\delta >0$ such that

$$d_X(x,y)<\delta ⟹d_Y(f(x),f(y))<ϵ.$$

Lemma (Munkres 21.2 & 21.3): Convergent sequence definition of continuity in metrizable spaces

1. Let $A\subseteq X$. If a sequence in $A$ converges to $x\in X$, then $x\in \overline{A}$, the closure of $A$.
2. Let $f:X\to Y$. If $f$ is continuous, then for all sequences $x_n\to x$ in $X$, we have $f(x_n)\to f(x)$ in $Y$.

The converse holds for metrizable spaces. As a corollary, The countably infinite product of the real line with itself in the box topology is not metrizable.

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Examples §

Basic examples of continuous functions in $\mathbb{R}$ §

More continuous functions can be generated from the following rules:

• The identity function $f(x)=x$ is continuous.
• Constant functions $f(x)=c$ for some chosen $c\in \mathbb{R}$ are continuous.
• The function $f(x)=1/x$ is continuous on $\mathbb{R}\backslash \{0\}$.
• The function $f:(0,\infty )\to \mathbb{R}$ defined by $f(x)=\sqrt{x}$ is continuous.
• Products, sums, and composites of continuous functions (when defined) are continuous.

Continuity vs. uniform continuity §

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Review §

Topology

• Show that for $f:\mathbb{R}\to \mathbb{R}$, the $ϵ-\delta$ definition and the definition of a continuous function in topological space are equivalent. ⭐
• Prove the equivalence of statements in Theorem (Munkres 18.1). ⭐
• Prove the pasting lemma. ⭐
• Prove that the $ϵ-\delta$ definition of continuity holds for general metric spaces (i.e., with a metric topology).

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Proof appendix §

Lemma (Munkres 21.2 & 21.3): Convergent sequence definition of continuity in metrizable spaces

1. Let $A\subseteq X$. If a sequence in $A$ converges to $x\in X$, then $x\in \overline{A}$, the closure of $A$.
2. Let $f:X\to Y$. If $f$ is continuous, then for all sequences $x_n\to x$ in $X$, we have $f(x_n)\to f(x)$ in $Y$.

The converse holds for metrizable spaces. As a corollary, The countably infinite product of the real line with itself in the box topology is not metrizable.

Link to original

#wip