Basic Theorems Regarding Continuity of Functions on Metric Spaces

# Basic Theorems Regarding Continuity of Functions on Metric Spaces

Recall from the Continuity of Functions on Metric Spaces page that if $(S, d_S)$ and $(T, d_T)$ are two metric spaces and $p \in S$ then a function $f : S \to T$ is said to be continuous at $p$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $d_S(x, p) < \delta$ then:

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\begin{align} \quad d_T(f(x), f(p)) < \epsilon \end{align}

Furthermore, we said that $f$ is continuous on all of $A \subseteq S$ if $f$ is continuous at each point in $A$.

We will now look at some nice theorems regarding continuity of functions on metric spaces.

Theorem 1: Let $(\mathbb{R}^n, d)$ be a metric space on $\mathbb{R}^n$ and let $S \subseteq \mathbb{R}^n$ and $f : S \to \mathbb{R}$. If $\mathbf{p} \in S$, $f$ is continuous at $\mathbf{p}$, and $f(\mathbf{p}) > 0$ then there exists an open ball $B(\mathbf{p}, r)$ in $S$ such that $f(\mathbf{x}) > 0$ for all $\mathbf{x} \in B(\mathbf{p}, r)$. Similarly, if $f(\mathbf{p}) < 0$ then there exists an open ball $B(\mathbf{p}, r)$ in $S$ such that $f(\mathbf{x}) < 0$ for all $\mathbf{x} \in B(\mathbf{p}, r)$. |

**Proof:**Let $f$ is continuous at $\mathbf{p}$ and let $f(\mathbf{p}) > 0$. Since $f$ is continuous we have that the inverse image of any open set is open. In particular, $\mathbb{R}$ is open and so $f^{-1}(\mathbb{R}) = S$ is open.

- Since $S$ is open and $\mathbf{p} \in S$ we have that $\mathbf{p} \in \mathrm{int} (S)$ and so there exists an $r^* > 0$ such that:

\begin{align} \quad B(\mathbf{p}, r^*) \subseteq S \quad (*) \end{align}

- Furthermore, since $f(\mathbf{p}) >0$, then for $\epsilon_1 = \frac{f(\mathbf{p})}{2} > 0$ there exists a $\delta_1 > 0$ such that if $d(\mathbf{x}, \mathbf{p}) < \delta_1$ then:

\begin{align} \quad \mid f(\mathbf{x}) - f(\mathbf{p}) \mid < \epsilon_1 = \frac{f(\mathbf{p})}{2} \quad \Leftrightarrow \quad \frac{f(\mathbf{p})}{2} < f(\mathbf{x}) < \frac{3f(\mathbf{p})}{2} \end{align}

- In particular, for $\mathbf{x} \in B(\mathbf{p}, \delta_1)$ we have that:

\begin{align} \quad f(\mathbf{x}) > \frac{f(\mathbf{p})}{2} > 0 \quad (**) \end{align}

- So, take $r = \min \{ r^*, \delta_1 \}$. So the open ball $B(\mathbf{p}, r)$ is such that $(*)$ holds (so that this open ball is contained in $S$) and such that $(**)$ holds (so that every point in this open ball is mapped to the positive real numbers).

- An analogous argument can be made for when $f(\mathbf{p}) < 0$. $\blacksquare$