September 22th, 2026 · Real Analysis

[0.0.8]Definition(Limit of a Function)#

Let $A \subseteq \bR$ be some subset of $\bR$ , and consider a function $f: A \to \bR$ . For some limit point $a$ of $A$ , we say the limit of $f(x)$ as $x$ approaches $a$ is $L$ if, for any $\epsilon > 0$ , there exists a $\delta > 0$ such that

0 < |x - a| < \delta \implies |f(x) - L| < \epsilon

for all $x \in A$ . We write this as

\lim_{x \to a} f(x) = L.

[0.0.9]Definition(Continuity of a Function)#

Let $A \subseteq \bR$ be some subset of $\bR$ , and consider a function $f: A \to \bR$ . For some $a \in A$ , we say $f$ is continuous at $a$ if, for any $\epsilon > 0$ , there exists $\delta > 0$ such that

|x - a| < \delta \implies |f(x) - f(a)| < \epsilon

for all $x \in A$ .

[0.0.10]Definition(Derivative of a Function)#

Let $A \subseteq \bR$ be some subset of $\bR$ , and consider a function $f: A \to \bR$ . For some interior point $a \in A$ , provided the limit exists, we say the derivative of $f$ at $a$ is given by

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}.