Limits of Functions on Metric Spaces

We can define many different functions from subsets of the real numbers back into the real numbers - and now we will begin to look at functions defined on metric spaces.

We will use the notation $(S, d_S)$ and $(T, d_T)$ to denote metric spaces of the sets $S$ and $T$ with metrics $d_S : S \times S \to [0, \infty)$ and $d_T : T \times T \to [0, \infty)$ respectively. We first define what exactly a function from $(S, d_S)$ to $(T, d_T)$ is.

Definition: Let $(S, d_S)$ and $(T, d_T)$ be two metric spaces and let $A \subseteq S$. A Function from $(S, d_S)$ to $(T, d_T)$ is a rule $f : A \to T$ defined for all $x \in A$ by $f(x) = y$ for $y \in T$. The Domain of $f$ is the set $A$ denoted $D(f) = A$. The Codomain of $f$ is the set $T$ denoted $C(f) = T$. The Range of $f$ is a subset of $T$ denoted $R(f) \subseteq T$ and is the image of $A$ under $f$, i.e., $R(f) = f(A) = \{ f(x) = y \in T : x \in A \}$.

Like with limits of functions over the set of real numbers - we can also define limits of functions on metric spaces.

Definition: Let $(S, d_S)$ and $(T, d_T)$ be metric spaces, $A \subseteq S$, and $f : A \to T$. Furthermore, let $p \in S$ be an accumulation point of $A$ and let $b \in T$. Then we say that $\lim_{x \to p} f(x) = b$ if and only if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $x \in D(f) \setminus \{ p \}$ and $d_S(x, p) < \delta$ we have that $d_T(f(x), b) < \epsilon$.

It is important to note that $p \in S$ must be an accumulation point so that $x \to p$ makes sense, i.e., so that $x$ can approach $p$.

An equivalent definition is that $\lim_{x \to p} f(x) = b$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $x \in [D(f) \setminus \{ p \}] \cap B_S(p, \delta)$ then $f(x) \in B_T(b, \epsilon)$.

Notice that the definition above is a generalization of a limit of a function $f : A \to \mathbb{R}$ where $A \subseteq \mathbb{R}$ and the metric defined for the domain and codomain is the usual Euclidean metric $d(x, y) = \mid x - y \mid$. For $c$ as a cluster point (equivalently, an accumulation point) recall that $\lim_{x \to c} f(x) = A$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $x \in D(f) \setminus \{ c \}$ and $0 < d(x, c) = \mid x - c \mid < \delta$ then $d(f(x), A) = \mid f(x) - A \mid < \epsilon$.

Like with sequences in metric spaces - many of the proofs regarding functions on metric spaces are generalizations of that of functions from a subset of $\mathbb{R}$ to $\mathbb{R}$.