Exercise 4.6.6

Question

Construct a bijection between the set of jump discontinuities of a monotone function $f$ and a subset of $\mathbf{Q}$. Conclude that $D_{f}$ for a monotone function $f$ must either be finite or countable, but not uncountable.

Ulisse Mini · Accepted Answer

In 4.6.5 we showed every $c \in D_f$ is a jump discontinuity, i.e. both sided limits $L$ and $M$ exist and $L 
e M$. Pick some $r \in (L,M) \cap \mathbf{Q}$ and assign $f(c) = r$. Continue like this to define a bijection $f : D_f 	o Q$ where $Q \subseteq \mathbf{Q}$. Thus $D_f$ must be finite or countable.

Exercise 4.6.6

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Exercise 4.6.6

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