Characteristics of a monotonic function on a closed interval

Question

Let $f:[a,b]$ be a monotonic function. Then

1. $f$ is continuous.
2. $f$ is discontinuous at most two points.
3. $f$ is discontinuous at finitely many points.
4. $f$ is discontinuous at most countable points.

Solution:

• $f:[0,1]\rightarrow \mathbb R$ defined by $f(x)=x$ is monotonic and continuous.So,$(1)$ holds.

• $f:[-5,5]\rightarrow \mathbb R$ defined by $f(x)= \left\{ \begin{array}{c}0;x<-1\\ 1;-1\leq x<0 \\ 2;0\leq x<1 \\ 3;x\geq1 \end{array} \right. $ is montonic and discontinuous at three points.Hence,$(2)$ does not hold.

• $f:[0,1]\bigcap \mathbb Q^c\rightarrow \mathbb R$ defined by $f(x)=x$ is monotonic and discontinuous at uncountabe points.So,$(3)$ & (4)does not holds.

Please check my solution!

Answer 1

Consider the function

$$f(x):x\in\left(\frac1{n+1},\frac1n\right]\to\frac xn,\\0\to0.$$

It is monotonic in $[0,1]$ but has countably many discontinuities, hence it invalidates 1., 2. and 3.