Continuity on R

Subjects: Real Analysis
Links: Functional Limits in R, Open and Closed Sets in R

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

Definitions

$ε$ - $δ$ Definition

\forall ε > 0 \exists δ > 0 \forall x \in D [| x - c | < δ ⟹ | f (x) - f (c) | < ε]

Sequential Definition

\forall (x_{n})_{n \in N} \subseteq D [lim_{n \to \infty} x_{n} = c ⟹ lim_{n \to \infty} f (x_{n}) = f (c)]

Topological Definition

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

\forall ε > 0 \exists δ > 0 \forall x \in D [x \in V_{δ} (c) ⟹ f (x) \in V_{ε} (f (c))]

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 $lim_{n \to \infty} x_{n} = c$ , such that $lim_{n \to \infty} f (x_{n}) \neq f (c)$ . Then, $f$ is discontinuous at $c$ .

Algebraic properties

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

for any $α \in R, α f$ is continuous at $c$
$f + g$ is continuous at $c$
$f g$ is continuous at $c$

Composition of Continuous Functions

Given $f : A \to R$ , and $g : B \to R$ , 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:

lim_{x \to c} (g (f (x)) = g (lim_{x \to c} f (x))

Preservation of Compactness

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

Extreme Value Theorem

Let $f : K \to R$ , is continuous on $K \subseteq R$ , then $f$ attains a maximum and a minimum value. In other words there are $x_{1}$ and $x_{2}$ such that $f (x_{1}) \leq f (x) \leq f (x_{2})$ , for all $x \in K$

Continuity of the Inverse

Let $f : D \to R$ , 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