Continuity on R

$f:D\to \mathbb{R}$, let $c\in D$, $f$ is a continuous at $c$ iff:

Definitions

$\epsilon$-$\delta$ Definition

Sequential Definition

Topological Definition

Given a neighbourhood $V$ of $f(c)$, then there’s a neighbourhood $U$ of $c$, such that $f(U)\subseteq V$.

Let $O$ be an open subset relative to $f(D)$, $f$ is continuous on $D$ iff $f^{-1}(O)$ is open

Divergence Criterion

Let $(x_n)$ $\subseteq D$ such that $\underset{n\to \mathrm{\infty }}{lim}{x}_n=c$, such that $\underset{n\to \mathrm{\infty }}{lim}f(x_n)\ne f(c)$. Then, $f$ is discontinuous at $c$ .

Algebraic properties

Let $f,g:D\to \mathbb{R}$, where $f$ and $g$ are continuous at $c$ then:

1. for any $\alpha \in \mathbb{R},\alpha f$ is continuous at $c$
2. $f+g$ is continuous at $c$
3. $fg$ is continuous at $c$

Composition of Continuous Functions

Given $f:A\to \mathbb{R}$, and $g:B\to \mathbb{R}$, assume $f(A)\subseteq B$, so that $g\circ f$ is well defined. Let $f$ be continuous at $c$, and $g$ be continuous at $f(c)$, then $g\circ f$ is continuous at $c.$

Corollary: If $g$ is continuous, then:

Preservation of Compactness

Let $f:A\to \mathbb{R}$, where $K$ be a compact subset of $A$, then $f(K)$ is also compact.

Extreme Value Theorem

Let $f:K\to \mathbb{R}$, is continuous on $K\subseteq \mathbb{R}$, then $f$ attains a maximum and a minimum value. In other words there are $x_1$ and $x_2$ such that $f(x_1)\le f(x)\le f(x_2)$, for all $x\in K$

Continuity of the Inverse

Let $f:D\to \mathbb{R}$, is continuous and injective, defining $f^{-1}$ over the $f(D)$ in the natural way: $f(x)=y$ iff $f^{-1}(y)=x$, then $f^{-1}$ is continuous on $f(D)$.

We have stronger version like Uniform Continuity on R

We can also explore the Discontinuities on R and how they behave