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Wilson A. Sutherland - Introduction to Metric and Topological Spaces-Oxford University Press (2009)
Exercise 4.18.★★ For any function $f:\Bbb R→\Bbb R$ show that the set of points $a∈\Bbb R$ at which $f$ has a simple jump discontinuity is countable.
S.4.pdf
4.18 There are two ways in which $f$ can have a simple jump discontinuity at $a \in \mathbb{R}$ : either the left- and right-hand limits of $f$ at $a$ differ, or they are equal but differ from $f(a)$. One can show that each of the sets at which these two possibilities occur is countable.
To deal with the first kind of point, for each $n \in \mathbb{N}$ let $D_n$ be the set of points $a \in \mathbb{R}$ such that $\left|\lim _{x \rightarrow a-} f(x)-\lim _{x \rightarrow a+} f(x)\right| \geqslant 1 / n$. Then the points where the left- and right-hand limits of $f$ differ is the countable union of $D_n$ over $n \in \mathbb{N}$. With some effort one can prove that $D_n$ is countable (the key is to show that if $a \in D_n$ then there exists $\delta_a>0$ such that there is no other point of $D_n$ within $\delta_a$ of $a$. Then we can choose a rational number in each interval $\left(a, a+\delta_a\right)$, get an injective function from $D_n$ to $\mathbb{Q}$, and apply Corollary S.2.7 to show that $D_n$ is countable.) Since a countable union of countable sets is countable, the first kind of jump discontinuities form a countable set.
The proof that the other kind of discontinuities form a countable set is similar, replacing $\left|\lim _{x \rightarrow a-} f(x)-\lim _{x \rightarrow a+} f(x)\right|$ by $\left|f(a)-\lim _{x \rightarrow a+} f(x)\right|$.
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