[4.2.1] Definition: Functional Limit
Let \( f: A \rightarrow \mathbb{R} \) and let \( c \) be a limit point of \( A \). We say that
\[
\lim_{x \to c} f(x) = L
\]
if for all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \) (and \( x \in A \)), it follows that
\[
|f(x) - L| < \epsilon
\]
Recall the definition of a limit point. So as \(x\) gets closer to the limit point \(c\), the value of the function at \(x\) will get closer to \(L\).
[4.2.1B] Definition: Functional Limit: The Topological Version
Let \( f: A \rightarrow \mathbb{R} \) and let \( c \) be a limit point of \( A \). We say that
\[
\lim_{x \to c} f(x) = L
\]
if for every \(\epsilon\)-neighborhood \( V_{\epsilon} \) of \( L \), there exists a \(\epsilon\)-neighborhood \( V_{\epsilon}(c) \) around \( c \) with the property that for all \( V_{\epsilon}(c) \) different from \( c \) (with \( x \in A \)) it follows that \( f(x) \in V_{\epsilon}(L) \).