Continuous Functions on Metric Spaces Review

# Continuous Functions on Metric Spaces Review

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

Let $(S, d_S)$ and $(T, d_T)$ be metric spaces and let $f : S \to T$.

- On the
**Continuity of Functions on Metric Spaces**page we said that $f$ is continuous at $p \in S$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $d_S(x, p) < \delta$ then $d_T(f(x), f(p)) < \epsilon$.

- Equivalently, $f$ is continuous at $p$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that the image of the ball centered at $p$ with radius $\delta$ (in $S$) is fully contained in the ball centered at $f(p)$ with radius $\epsilon$ (in $T$), i.e.:

\begin{align} \quad f(B_S(p, \delta)) \subseteq B_T(f(p), \epsilon) \end{align}

- We noted that if $p$ is an accumulation point of $S$ then $f$ is continuous at $p$ if and only if:

\begin{align} \quad \lim_{x \to p} f(x) = f(p) \end{align}

- On the other hand, if $p$ is an isolated point of $S$ then $f$ is always continuous at $p$ since there will exist a $\delta > 0$ such that $B_S(p, \delta)$ contains only $p$, and so $f(B_S(p, \delta)) = \{ f(p) \}$ and $\{ f(p) \} \subseteq B_T(f(p), \epsilon)$ for all $\epsilon > 0$.

- On the
**Sequential Criterion for the Continuity of a Function on Metric Spaces**page we looked at very important criterion for the continuity of a function on metric spaces at a point. We said that $f$ is continuous at $p$ if and only if for all sequences $(x_n)_{n=1}^{\infty}$ in $S$ that converge to $p$ we have that the sequences $(f(x_n))_{n=1}^{\infty}$ in $T$ converge to $f(p)$.

- On
**The Continuity of Composite Functions on Metric Spaces**page we further showed that if $(S, d_S)$, $(T, d_T)$, and $(U, d_U)$ are all metric spaces with $f : S \to T$ and $g : T \to U$ then if $f$ is continuous at $p \in S$ and $g$ is continuous at $f(p) \in T$ then the composite function $g \circ f : S \to U$ is continuous at $p$.

- We then turned out attention towards the continuity of complex-valued functions. On the
**Continuity of Combinations of Complex-Valued Functions**page we said that if $(S, d_S)$ and $(\mathbb{C}, d)$ are metric spaces where $d(x, y) = \mid x - y \mid$ for all $x, y \in \mathbb{C}$, and if $f, g : S \to \mathbb{C}$, $p \in S$, and $c$ is a constant then if $f$ and $g$ are continuous at $p$ then $f + g$, $f - g$, $cf$ (for any $c, and [[$ fg$ are continuous at $p$.

- We then looked at the continuity of vector-valued functions. On the
**Continuity of Combinations of Vector-Valued Functions**page we said that if $(S, d_S)$ and $(\mathbb{R}^n, d)$ are metric spaces where $d(\mathbf{x}, \mathbf{y}) = \| \mathbf{x} - \mathbf{y} \|$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, and if $\mathbf{f}, \mathbf{g} : S \to \mathbb{R}^n$, $p \in S$, and $\lambda$ is a constant then if $\mathbf{f}$ and $\mathbf{g}$ are continuous at $p$ then $\mathbf{f} + \mathbf{g}$, $\mathbf{f} - \mathbf{g}$, $\lambda \mathbf{f}$, $\mathbf{f} \cdot \mathbf{g}$, and $\| \mathbf{f} \|$ are all continuous at $p$.

- On the
**Open and Closed Set Criteria for Continuity of Functions on Metric Spaces**page we looked at a nice characterization for the continuity of a function. We saw that $f$ is continuous on $S$ if and only if for every open set $V$ in $T$ we have that the inverse image $f^{-1}(V)$ is open in $S$.

- Similarly, $f$ is continuous on $S$ if and only if for every closed set $V$ in $T$ we have that the inverse image $f^{-1}(V)$ is closed in $S$.

- We then looked at a very nice property of continuous functions with compact domains. On the
**Continuous Functions on Compact Sets of Metric Spaces**page. We saw that if $f$ is continuous, $X \subseteq S$, and $X$ is compact in $S$ then the image $f(X)$ is compact in $T$, and as a corollary, $f(X)$ will be closed, bounded, and every infinite subset of $f(X)$ will have an accumulation point.

- On the
**The Extreme Value Theorem for Continuous Functions on Compact Sets of Metric Spaces**page we looked at a generalization of the famous Extreme Value theorem. We said that $(S, d_S)$ and $(\mathbb{R}, d)$ are metric spaces where $d(x, y) = \mid x - y \mid$ for all $x, y \in \mathbb{R}$, and if $X \subseteq S$ is compact in $S$ and $f : S \to T$ is continuous then $f(X)$ is closed and bounded in $T$ and achieves its supremum and infimum on $X$, that is, there exists $p, q \in X$ such that:

\begin{align} \quad f(p) = \sup \{ f(x) : x \in X \} \quad, \quad f(q) = \inf \{ f(x) : x \in X \} \end{align}