91Ó°ÊÓ

A continuous function is a fundamental concept in mathematics and analysis. To understand continuity, imagine a function that turns small changes in the input into small changes in the output. Formally, a function \( f : X \rightarrow Y \) is continuous at a point \( x \) in \( X \) if, for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( d_X(x, x') < \delta \), we have \( d_Y(f(x), f(x')) < \varepsilon \).
Essentially, this means that even a tiny wiggle at \( x \) won’t cause a huge change in \( f(x) \). If a function is continuous at every point in a space, it's simply called continuous. This seamless behavior helps ensure predictability and smoothness of functions.

Metric spaces are structured sets that allow us to discuss the concept of distance. A metric space \( (X, d) \) consists of a set \( X \) and a distance function \( d \) that satisfies a few conditions:

• Non-negativity: \( d(x, y) \geq 0 \) for all \( x, y \in X \).
• Identity of indiscernibles: \( d(x, y) = 0 \) if and only if \( x = y \).
• Symmetry: \( d(x, y) = d(y, x) \) for all \( x, y \in X \).
• Triangle inequality: \( d(x, z) \leq d(x, y) + d(y, z) \) for all \( x, y, z \in X \).

These conditions ensure a well-defined concept of distance within the set. Metric spaces are central to many areas in mathematics because they provide a rigorous way to discuss limits, convergence, and continuity.

A \( G_{\delta} \) set is a special type of set in topology, especially useful in metric spaces. A \( G_{\delta} \) set is constructed as a countable intersection of open sets. To break this down:

• Open Sets: These are sets where, for any point within them, there's always a little room (or neighborhood) around that point still within the set.
• Countable Intersection: This means we're taking an intersection of a sequence (which can go on forever but is countable) of such open sets.

In the context of the problem, we denote \( S \) as the set of points in \( X \) where the function \( f \) is continuous. We then express \( S \) as a countable intersection of open sets \( U_{n,m} \), defined by pairs of natural numbers \( n \) and \( m \). For each pair, an open set \( U_{n,m} \) comprises points where \( f \) is approximately continuous (within a precision dictated by \( \varepsilon = \frac{1}{n} \) and \( \delta = \frac{1}{m} \)). This means:

\[U_{n,m} = \{ x \in X \mid \forall x' \in B(x, \frac{1}{m}), d_Y(f(x), f(x')) < \frac{1}{n} \}\]

Thus, \( S \) can be written as a \( G_{\delta} \) set:

\[S = \bigcap_{n=1}^{\infty} \bigcap_{m=1}^{\infty} U_{n,m}\]

This formulation as a countable intersection of open sets confirms \( S \) is indeed a \( G_{\delta} \) set.