Let (Ω1, d1) and (Ω2, d2) be metric spaces and let f: Ω1 →Ω2 be an arbitrary map. Denote by Uf={

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Let (Ω1, d1) and (Ω2, d2) be metric spaces and let f: Ω1 →Ω2...

Let $\left(\Omega_{1}, d_{1}\right)$ and $\left(\Omega_{2}, d_{2}\right)$ be metric spaces and let $f: \Omega_{1} \rightarrow \Omega_{2}$ be an arbitrary map. Denote by $U_{f}=\left\{x \in \Omega_{1}: f\right.$ is discontinuous at $\left.x\right\}$ the set of points of discontinuity of $f$. Show that $U_{f} \in \mathcal{B}\left(\Omega_{1}\right)$. Hint: First show that for any $\varepsilon>0$ and $\delta>0$ the set $$ U_{f}^{\delta, \varepsilon}:=\left\{x \in \Omega_{1}: \text { there are } y, z \in B_{\varepsilon}(x) \text { with } d_{2}(f(y), f(z))>\delta\right\} $$ is open (where $\left.B_{\varepsilon}(x)=\left\{y \in \Omega_{1}: d_{1}(x, y)<\varepsilon\right\}\right) .$ Then construct $U_{f}$ from such $U_{f}^{\delta, \varepsilon}$

Let $\left(\Omega_{1}, d_{1}\right)$ and $\left(\Omega_{2}, d_{2}\right)$ be metric spaces and let $f: \Omega_{1} \rightarrow \Omega_{2}$ be an arbitrary map. Denote by $U_{f}=\left\{x \in \Omega_{1}: f\right.$ is discontinuous at $\left.x\right\}$ the set of points of discontinuity of $f$. Show that $U_{f} \in \mathcal{B}\left(\Omega_{1}\right)$.
Hint: First show that for any $\varepsilon>0$ and $\delta>0$ the set
$$
U_{f}^{\delta, \varepsilon}:=\left\{x \in \Omega_{1}: \text { there are } y, z \in B_{\varepsilon}(x) \text { with } d_{2}(f(y), f(z))>\delta\right\}
$$
is open (where $\left.B_{\varepsilon}(x)=\left\{y \in \Omega_{1}: d_{1}(x, y)<\varepsilon\right\}\right) .$ Then construct $U_{f}$ from such $U_{f}^{\delta, \varepsilon}$

Key Concepts

Set Construction Techniques

Constructing the set of discontinuities as a countable union and intersection of open sets relies on standard techniques in topology and measure theory. By expressing the set of discontinuity points as a combination of open sets generated by conditions derived from the epsilon-delta definitions, one shows such a set is a Borel set, which is crucial for further analysis in measurable spaces.

Open Sets and the Epsilon-Delta Condition

Open sets in metric spaces are defined by the presence of a neighborhood around each of their points, meaning each point has an ?-ball completely contained in the set. By using the epsilon-delta definition of continuity, one can characterize certain collections of points (such as those where a function behaves erratically) as being open, by constructing appropriate balls around the points in question.

Borel Sigma Algebra

The Borel sigma algebra of a metric space is the smallest sigma algebra containing all open sets. This concept is essential in measure theory and probability, as it allows one to define measurable functions and sets, and ensures that common operations on open sets (such as countable unions and intersections) are included.

Continuity in Metric Spaces

Continuity of a function between metric spaces is defined in an epsilon-delta framework: a function is continuous at a point if for every ? > 0, there exists a ? > 0 such that whenever two points in the domain are within ? of the point, their images are within ? of each other. This concept is crucial for understanding the behavior of functions and their limits.

Metric Spaces

A metric space is a set equipped with a metric, a function that assigns a non-negative real number to each pair of points, satisfying properties like non-negativity, symmetry, and the triangle inequality. This structure facilitates the discussion of concepts like distance, convergence, and neighborhood, which are fundamental in analysis and topology.

Discontinuity Points

Discontinuity points of a function are those points in the domain where the function fails to be continuous. Identifying these points is important in analysis, as it helps in understanding the overall structure and behavior of the function, and in constructing appropriate sets that capture irregular behavior.

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