Continuity of the distance to a point or a set

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Problem 4.2.7 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 4.2 "General Theory"

Let $M$ be a metric space with metric $d$ . Let $C$ be a nonempty closed subset of $M$ . Define $f : M \to \mathbb {R}$ by

f(x) = \inf \left\{ d(x,y) \mid y \in C\right\} .

Show that $f$ is continuous, and that $f(x) = 0$ if and only if $x \in C$ .

Problem 2.3.37 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 2.3 "Compactness"

Let ( $M, d$ ) be a metric space, let $A \subset M$ , and let $h(x)=d(x, A)= \inf \left\{ d(x, a): a \in A\right\}$ . Prove that $d$ is a continuous function on $M$ .

Problem 2.4.27 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 2.4 "Limits of Functions"

Let ( $M, d$ ) be a metric space and $x_{0} \in M$ . Let $f: M \rightarrow \mathbb {R}$ be defined by $f(x)=d\left(x, x_{0}\right)$ . Prove that $f$ is uniformly continuous.

Problem 13.26 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 13.3 "Continuous Mappings Between Metric Spaces"

Let $z$ be a point in the metric space ( $X, \rho$ ). Define the function $f: X \rightarrow \mathbb {R}$ by $f(x) = \rho (x, z)$ . Show that $f$ is uniformly continuous.

Problem 13.27 from Royden H.L., Fitzpatrick P.M. "Real Analysis" [RoFi23]More problems in Section 13.3 "Continuous Mappings Between Metric Spaces"

(i)(ii)

For a non-empty subset $E$ of the metric space ( $X, \rho$ ) and a point $x \in X$ , define the distance from $x$ to $E, \operatorname {dist}_{E}(x)$ , as follows:

\operatorname {dist}_{E}(x) = \inf \left\{ \rho (x, y) \mid y \in E\right\} .

(i)

Show that $\left|\operatorname {dist}_{E}(x)-\operatorname {dist}_{E}(y)\right| \leq \rho (x, y)$ for all $x, y \in X$ .

(ii)

Show that $\left\{ x \in X \mid \operatorname {dist}_{E}(x) = 0\right\} = \bar{E}$ .

Problem 1.4.20 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.4 "Continuity"

(a)(b)

If $E$ is a nonempty subset of a metric space $X$ , define the distance from $x \in X$ to $E$ by

\rho _{E}(x) = \inf _{z \in E} d(x, z)

(a)

Prove that $\rho_{E}(x) = 0$ if and only if $x \in E$ .

(b)

Prove that $\rho_{E}$ is a uniformly continuous function on $X$ , by showing that

\left|\rho _{E}(x)-\rho _{E}(y)\right| \leq d(x, y)

for all $x \in X, y \in X$ . Hint: $\rho_{E}(x) \leq d(x, z) \leq d(x, y)+d(y, z)$ , so that

\rho _{E}(x) \leq d(x, y)+\rho _{E}(y)

Problem 5.15 from Schilling R.L. "Brownian Motion: A Guide to Random Processes and Stochastic Calculus" [Sch21]More problems in Chapter 5 "Brownian motion as a martingale"

Show that the function $d(x, A): = \inf_{y \in A}\left|x - y\right|, A \subset \mathbb {R}^{d}$ , is continuous.

Problem 7.8 from Schilling R.L. "Brownian Motion: A Guide to Random Processes and Stochastic Calculus" [Sch21]More problems in Chapter 7 "Brownian motion and transition semigroups"

(a)(b)(c)

(a)

Let $d(x, A): = \inf_{a \in A}\left|x - a\right|$ be the distance between the point $x \in \mathbb {R}^{d}$ and the set $A \in \mathscr {B}\left(\mathbb {R}^{d}\right)$ . Show that $x \mapsto d(x, A)$ is continuous.

(b)

Let $u_{n}$ be as in (7.7). Show that $u_{n} \in \mathcal{C}_{c}\left(\mathbb {R}^{d}\right)$ and $\inf_{n} u_{n}(x) = \; \mathbb {1}_{K}(x)$ . Draw a picture if $d = 1$ and $K = [0,1]$ .

(c)

Let $\chi_{n}$ be a sequence of type $\delta$ , i.e. $\chi_{n} \in \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right)$ , $\operatorname {supp} \chi_{n} \subset \mathbb {B}(0,1 / n), \chi_{n} \geqslant 0$ and $\int \chi_{n}(x) d x = 1$ . Show that $v_{n}: = \chi_{n} \star u_{n} \in \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right)$ are smooth functions such that $\lim_{n \rightarrow \infty } v_{n} = \; \mathbb {1}_{K}$ .

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[Pet25] Petrovic J.S. Measure and Integral: Theory and Practice. CRC Press, 2025.

[RoFi23] Royden H.L., Fitzpatrick P.M. Real Analysis. 5th ed., Pearson, 2023.

[Rud76] Rudin W. Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.

[Sch21] Schilling R.L. Brownian Motion: A Guide to Random Processes and Stochastic Calculus. 3rd ed., De Gruyter, 2021.