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 Uniqueness of Limits of Functions on Metric Spaces page we saw that if the limit of $f$ as $x$ approaches $p$ exists then that limit is unique - as expected.

• 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 $T$.

• We then turned out attention to limits of complex-valued functions. On the Limits of Sums and Differences of Complex-Valued Functions, Limits of Products of Complex-Valued Functions, and the Limits of Reciprocals and Quotients of Complex-Valued Functions pages we looked at a bunch of theorems regarding the limits of complex-valued functions. We said that if $(S, d_S)$ and $(\mathbb{C}, d)$ are metric spaces where $d$ is the usual metric on $\mathbb{C}$ defined for all $x, y \in \mathbb{C}$ by $d(x, y) = \mid x - y \mid$, $A \subseteq S$, $p \in S$ is an accumulation point of $A$, and $a, b \in T$ with $\displaystyle{\lim_{x \to p} f(x) = a}$ and $\displaystyle{\lim_{x \to p} g(x) = b}$ then we can conclude that:

(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}

• We then turned our attention to vector-valued functions. On the Existence of Limits of Vector-Valued Functions, Limits of Sums and Differences of Vector-Valued Functions, Limits of Scalar Multiples of Vector-Valued Functions, Limits of Dot Products of Vector-Valued Functions, and Limits of Norms of Vector-Valued Functions pages we saw that if $(S, d_S)$ and $(\mathbb{R}^n, d)$ are metric spaces where $d$ is the usual metric on $\mathbb{R}^n$ given for all $\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \mathbb{R}^n$ by $d(\mathbf{x}, \mathbf{y}) = \| \mathbf{x} - \mathbf{y} \|$, $A \subseteq S$, $p \in S$ is an accumulation point of $A$, $\mathbf{a}, \mathbf{b} \in \mathbb{R}^n$, $\mathbf{f}, \mathbf{g} : A \to \mathbb{R}^n$, and $\lim_{x \to p} \mathbf{f}(x) = \mathbf{a}$, $\lim_{x \to p} \mathbf{g}(x) = \mathbf{b}$, then:

(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}