Limits of Functions on Metric Spaces Review

Limits of Functions on Metric Spaces Review

We will now review some of the recent material regarding limits of functions on metric spaces.

Let $(S, d_S)$ and $(T, d_T)$ be metric spaces with $A \subseteq S$.

• On the Limits of Functions on Metric Spaces page we that a Function from the metric spaces $S$ and $T$ is a rule $f : A \to T$ defined for all $x \in A$ by $f(x) = y$ for some $y \in T$. The set $A$ is called the Domain of $f$ often written $D(f) = A$ and the set $T$ is called the Codomain of $f$ and is often written $C(f) = T$. The image of $A$ under $f$, $f(A)$, is called the Range of $f$.
• Furthermore, if $f : A \to T$ and $p \in S$ is an accumulation point on $A$ and $b \in T$ then we say that the Limit of $f$ as $x$ Approaches $p$ is $b$ written $\displaystyle{\lim_{x \to p} f(x) = b}$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $x \in D(f) \setminus \{ p \} = A \setminus \{p \}$ and $d_S(x, p) < \delta$ then:
(1)
\begin{align} \quad d_T(f(x), b) < \epsilon \end{align}
• An equivalent definition is that for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $x \in [D(f) \setminus \{ p \}] \cap B_S(p, \delta)$ then $f(x) \in B_T(b, \epsilon)$.
• Regardless, it is important to note that we require $p \in S$ to be an accumulation point of $A$ because then every open ball centered at $p$ contains points in $A$ and so we can "approach" $p$ by taking $x$ sufficiently closed to $p$ in terms of the metric defined on $S$.
• On the Sequential Criterion for the Limit of a Function on Metric Spaces page we looked at a very useful theorem which told us that the limit of $f$ as $x$ approaches $p$ is equal to $b$ if and only if for every sequence $(x_n)_{n=1}^{\infty}$ in $A \setminus \{ p \}$ that converges to $p$ in $S$ we have that the sequence $(f(x_n))_{n=1}^{\infty}$ converges to $b$ in $T$.
(2)
\begin{align} \quad \lim_{x \to p} [f(x) + g(x)] = a + b \end{align}
(3)
\begin{align} \quad \lim_{x \to p} [f(x) - g(x)] = a - b \end{align}
(4)
\begin{align} \quad \lim_{x \to p} f(x)g(x) = ab \end{align}
(5)
\begin{align} \quad \lim_{x \to p} \frac{1}{g(x)} = \frac{1}{b} \quad \mathrm{if \:} b \neq 0 \end{align}
(6)
\begin{align} \quad \lim_{x \to p} \frac{f(x)}{g(x)} = \frac{a}{b} \quad \mathrm{if \:} b \neq 0 \end{align}
(7)
\begin{align} \quad \lim_{x \to p} \mathbf{f}(x) = \mathbf{a} \quad \mathrm{if \: and \: only \: if} \quad \lim_{x \to p} f_j(x) = a_j \: , \forall j \in \{1, 2, ..., n \} \end{align}
(8)
\begin{align} \quad \lim_{x \to p} [\mathbf{f}(x) + \mathbf{g}(x)] = \mathbf{a} + \mathbf{b} \end{align}
(9)
\begin{align} \quad \lim_{x \to p} [\mathbf{f}(x) - \mathbf{g}(x)] = \mathbf{a} - \mathbf{b} \end{align}
(10)
\begin{align} \quad \lim_{x \to p} \lambda \mathbf{f}(x) = \lambda \mathbf{a} \:, \forall \lambda \in \mathbb{R} \end{align}
(11)
\begin{align} \quad \lim_{x \to p} \mathbf{f}(x) \cdot \mathbf{g}(x) = \mathbf{a} \cdot \mathbf{b} \end{align}
(12)
\begin{align} \quad \lim_{x \to p} \| \mathbf{f}(x) \| = \| \mathbf{a} \| \end{align}