Does a function with left limits have only countable jumps?

Asked Apr 16 '16 at 21:16

Active Apr 17 '16 at 22:35

Viewed 137 times

If $f:\mathbb R\to\mathbb R$ has left limits, does it have at most countable jumps? By the set of jumps of $f$ I mean the set $\{x\in\mathbb R:f(x^-)\neq f(x)\}$

This is indeed true if we add the hypothesis of having also right limits as it was proved in How much a càdlàg (i.e., right-continuous with left limits) function can jump? But the proof does use this additional hypothesis.

Any ideas?

edited Apr 13 '17 at 12:21

Community

asked Apr 16 '16 at 21:16

webbster

It can have non-jump discontinuities. – GEdgar Apr 16 '16 at 21:19
@GEdgar Yes, of course. My question is whether ${x\in \mathbb R: f(x^-)\neq f(x)}$ is countable. I'll edit my question to make this clearer. Thank you – webbster Apr 17 '16 at 22:31

Does a function with left limits have only countable jumps?

0 Answers0