Continuity

A function $f:D\to \mathbb{R}$ with $D\subseteq \mathbb{R}$ is continuous at a point $x_0\in D$ if small changes in the input lead to arbitrarily small changes in the output.

Definition

Let $f:D\to \mathbb{R}$ and $x_0\in D$. Then $f$ is continuous at $x_0$ if

$$\forall \epsilon >0\exists \delta >0\forall x\in D:|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon$$

This means that for every prescribed output tolerance $\epsilon$, we can choose an input tolerance $\delta$ such that all points $x$ sufficiently close to $x_0$ are mapped to values sufficiently close to $f(x_0)$

Definition

Let $f:D\to \mathbb{R}$. Then $f$ is continuous on $D$ if

$$\forall x_0\in D\forall \epsilon >0\exists \delta >0\forall x\in D:|x-x_0|<\delta \Rightarrow |f(x)-f(x_0)|<\epsilon$$

Sequential characterization

The function $f:D\to \mathbb{R}$ is continuous at $x_0\in D$ if and only if for every sequence $(x_n{)}_{n\ge 1}$ in $D$ with $x_n\to x_0$, we have

$$f(x_n)\to f(x_0).$$

The definition of convergence of sequences has the same logical structure: the index $n$ plays the role of the input variable, and $n\to \infty$ is the discrete analogue of a function limit. In this sense, sequence convergence is a special case of the general idea of limits.

Limits

Limit at $+\infty$

We write

$$\underset{x\to +\infty }{\lim }f(x)=L$$

if

$$\forall \epsilon >0\exists R>0:\forall x\in D:x>R⟹|f(x)-L|<\epsilon$$

Right-hand limit

We write

$$\underset{x\to x_0^{+}}{\lim }f(x)=L$$

if

$$\forall \epsilon >0\exists \delta >0:\forall x\in D\cap (x_0,x_0+\delta ):|f(x)-L|<\epsilon$$

Here $x$ approaches $x_0$ only from the right.