0

I'm trying to understand a solution to the following problem from Elon Lima's Real Analysis.

Prove that the set of discontinuity points of a monotone function is countable.

The proof goes like this. Let $D$ be the set of points at which a monotone function $f$ is discontinuous. For each $x \in D$, take $a_x = \min\{\lim_{y \to x^-}f(y), f(y), \lim_{y \to x^+}f(y) \}$ and $b_x = \max\{\lim_{y \to x^-}f(y), f(y), \lim_{y \to x^+}f(y) \}$. Then $a_x < b_x$ and you can choose a rational number $r(x) \in ]a_x, b_x[$, which gives the injection $x \mapsto ]a_x, b_x[ \mapsto r(x)$.

My question is: why do the lateral limits exist?

hdecristo
  • 747

1 Answers1

1

It is a well-known theorem that a monotone function has both a left hand and a right hand limit at each point of its domain. See e.g. Prove that right hand limit exist for monotone function and Show that one-sided limits always exist for a monotone function (on an interval).

Paul Frost
  • 87,968