monotone function on open interval is continuous except possibly at a countable number of points Here is the proof from Royden's book. 


  Assume $f$ is increasing. Furthermore, assume $(a,b)$ is bounded and
  $f$ is increasing on the closed interval $[a,b]$. Otherwise, express
  $(a,b)$ as the union of an ascending sequence of open, bounded
  intervals, the closures of which are contained in $(a,b)$, and take
  the union of the discontinuities in each of this countable collection
  of intervals. For each $x_0 \in (a,b)$, $f$ has a limit from the left
  and from the right at $x_0$. Define $$f(x_0^-) = \lim_{x \to x_0^-}
 f(x) = sup \{f(x)| a &lt; x &lt; x_0\}, $$ $$f(x_0^+) = \lim_{x \to x_0^+}
 f(x) = sup \{f(x)| x_0 &lt; x &lt; b\}. $$Since $f$ is increasing, $f(x_0^-)
 \leq f(x_0^+)$. The function $f$ fails to be continuous at $x_0$ if
  and only if $f(x_0^-) &lt; f(x_0^+)$, in which case we define the open
  "jump" interval $J(x_0)$ by $$J(x_0) = \{y|f(x_0^-) &lt; y &lt;
 f(x_0^+)\}.$$ Each jump interval is contained in the bounded interval
  $[f(a), f(b)]$ and the collection of jump intervals is disjoint.
  Therefore, for each natural number $n$, there are only a finite number
  of jump intervals of length greater than $1/n$. Thus the set of points
  of discontinuity of $f$ is the union of a countable collection of
  finite sets and therefore is countable.


It seems the book is assuming that $[f(a), f(b)]$ being finite, then what if it is not finite? i.e., $[f(a), f(b)]$ is of the form $[f(a), \infty]$. Can I write $[f(a), \infty] = \bigcup_{k=1}^\infty [f(a), k]$ and say for each number n, there are only countably sum of finite jump intervals of length greater than $\frac{1}{n}$ in each $[f(a), k]$, thus there are countably many jump intervals in $[f(a), \infty]$.
But how come there is only a finite number of jump intervals, and that too of the length provided above. I didn't get that.
But how come there is only a finite number of jump intervals, and that too of the length provided above. I didn't get that.