continuity of monotonically increasing function

Question

Let $f:[0,1]\to[0,1]$ be monotonically increasing. Which of the following statements is/are true?

$f$ must be continuous at all but finitely many points in $[0,1]$
$f$ must be continuous at all but countably many points in $[0,1]$
$f$ must be Riemann integrable
$f$ must be Lebesgue integrable

I know that set of discontinuities of monotonic function is at most countable so 1 is not true.

If the set of discontinuities is at most countable then how to conclude for the second option that is set of continuities of monotone function. Here domain is [0,1] .So the set of continuities can be uncountable?

Monotone function on $[a,b]$ is Riemann integrable and hence Lebesgue integrable hence option 3 and 4 are correct.

A monotonic function can only have discontinuity of the first kind — Li Chun Min, May 12 '17 at 08:57
$f$ is Riemann integrable iff $f$ is continuous almost everywhere on $[0,1]$. Therefore since a monotone function is Riemann integrable 2 must be correct. — Janitha357, May 12 '17 at 09:00

score 2 · Answer 1 · answered May 12 '17 at 09:06

2

Let $f(x+)=\lim_{y\rightarrow x^{+}}f(y)$ and $f(x-)=\lim_{y\rightarrow x^{-}}f(y)$. Note that $f(x+)\geq f(x-)~\forall x$ as $f$ is increasing. Then let $D_{n}=\{x:f(x+)-f(x-)\geq\frac{1}{n}\}$. As $f:[0,1]\rightarrow[0,1]$, hence $|D_n|\leq n$. Also the set of all discontinuities, $D=\cup_{n=1}^{\infty}D_n$, which is a countable union of finite sets, and hence is countable.

Also Lebesgue measure of a countable set is $0$. Hence Lebesgue measurable.

answered May 12 '17 at 09:06

QED

12,644

second option is correct? – dipali mali May 13 '17 at 11:17
yes it is correct – QED May 13 '17 at 11:57

continuity of monotonically increasing function

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