Pointwise continuity

Overview §

Continuity is the simplest restriction or analyzing functions.

The distance between outputs of a function, or error, is a very small value $ϵ>0$, and the distance between “close enough” inputs is a value $\delta (ϵ)>0$. What is precisely “close enough” depends on choice of $ϵ$.

See also: (Theorem) Intermediate value

---

Continuity in $\mathbb{R}$ §

Most presentations of continuity use the “epsilon-delta” property.

Definition: Pointwise continuity

Let $D\subset \mathbb{R}$ be a subset, and $f:D\to \mathbb{R}$ be any function. We say $f$ is continuous at $x_0\in D$ if for all $ϵ>0$, there exists a $\delta >0$—where the value of $\delta (x_0,ϵ)$ depends on both $x_0$ and $ϵ$—so that for all $x\in D$ with $|x-x_0|<\delta$ we have $|f(x)-f(x_0)|<ϵ$.

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

---

Epsilon-delta arguments §

• Problem: Show that some function $f:D\to \mathbb{R}$ is continuous.
• Proof: Given $x_0\in D$ with particular properties and a positive value $ϵ>0$, find $\delta >0$ such that $|x-x_0|<\delta ⟹|f(x)-f(x_0)|<ϵ$.
  1. State what you want to prove.
  2. Choose $\delta$ in terms of $ϵ,x_0$ (hardest step).
  3. Assume $|x-x_0|<\delta$.
  4. Justify choice of $\delta$ by doing the manipulations to show $|f(x)-f(x_0)|<ϵ$.
• Thought process:
  1. Begin with the thing to prove, the right-hand side of the implication. Try to rearrange it to isolate the expression $|x-x_0|$.
  2. If the expression for $ϵ$ still depends on $x$, notice what $\delta$ has to be. Choose $\delta$, then make a claim about what $|x|$ is the upper or lower bound for.
  3. Get an upper or lower bound for the expression with $x$ in terms of just $\delta ,x_0$. Often, this uses the Triangle inequality.
  4. Choose $\delta$ which satisfies this, use arithmetic to rewrite as an expression where $\delta$ depends on $x_0,ϵ$.

---

Continuity in metric spaces §

Definition: Pointwise continuity in metric spaces

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

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

We can also characterize pointwise continuity using neighborhoods:

$$f(B(a,\delta ))\subset B(f(a),ϵ)\text{and}B(a,\delta )\subset f^{-1}(B(f(a),ϵ)).$$

Finally, $f$ is continuous if it is continuous for all points $a\in X$.

---

Notes §

• A function $f:D\to \mathbb{R}$ is continuous at an input $x_0$ in the arbitrary subset $D$ if no matter how close the outputs are chosen to be, we can look at inputs close enough to $x_0$ so that inputs $\delta$-close to $x_0$ produce outputs $ϵ$-close to $f(x_0)$.