Notes - Analysis II HT23, Continuity

Let $f : E \to \mathbb{R}$, and $p \in E$. What is the epsilon-delta definition of $f$ being continuous at $p$?

\[\forall \epsilon > 0 \text{ } \exists \delta > 0 \text{ s.t. } \forall x \in E \text { } (|x-p| < \delta \implies |f(x) - f(p)|<\epsilon)\]

Let $f : E \to \mathbb{R}$ where $E \subseteq \mathbb R$. What useful result links continuity and limits?

$f$ is continuous at any isolated point of $E$.
If $p \in E$ is a limit point of $E$, then $f$ is continuous at $p$ if and only if $\lim _ {x\to p} f(x)$ exists and $\lim _ {x\to p} f(x) = f(p)$.

Let $f : E \to \mathbb{R}$ where $E \subseteq \mathbb{R}$. What useful result links continuity and sequences?

$f$ is continuous at $p$ if and only if for all sequences $(p _ n)$ where $p _ n \in E$, $p _ n \to p$ then $f(p _ n) \to f(p)$.

If $f, g$ are continuous, what is

\[\lim_{x\to p} g(f(x))\]

equal to?

\[g\left(\lim_{x \to p} f(x)\right)\]

What restriction of $f : E \to \mathbb{R}$ do you consider for right-continuity at $p \in E$?

\[f_{\big|E\cap [p, \infty)}\]