I don't really get how the proof is complete. So we can associate each discontinuity with a rational that is in-between the left-hand limit and the right-hand limit. But what if the left-hand and the right-hand limit are the same and is an irrational number?
About the cardinality of set of discontinuities of a monotone function on (a, b)
by Shamiur Rahman Ramim -
Number of replies: 3
In reply to Shamiur Rahman Ramim
Sv: About the cardinality of set of discontinuities of a monotone function on (a, b)
by Jacob Kuhlin -
What does the function look like at x if the left-and-right-hand limits are equal at x?
In reply to Jacob Kuhlin
Sv: About the cardinality of set of discontinuities of a monotone function on (a, b)
by Shamiur Rahman Ramim -
Hmm. I see now that the example i was thinking of is not monotonic. Assuming that f(x-) = f(+), and assuming that f is monotone, am I right in assuming that f(x-) \leq f(x) \leq f(x+) which would give us that there is no discontinuity?
In reply to Shamiur Rahman Ramim
Sv: About the cardinality of set of discontinuities of a monotone function on (a, b)
by Jacob Kuhlin -
Yes, exactly.