If $f$ is an increasing function defined on $[a,b]$ . Then show that the set of discontinuities of $f$ is countable and that $f$ has points of continuity in every open subinterval of $[a,b]$
Attempt: I have trouble understanding why the number of discontinuities of $f$ should be countable. For example, if we define a function which is increasing but different at each point in the interval $[0,1]$, then, there can be uncountable number of discontinuities.
I think if we prove that the number of discontinuities is countable, then we can prove that the function has points of continuity in every open interval of $[a,b]$.
Please help me understand intuitively why $f$ must necessarily have countable number of discontinuities. Suppose, $f$ has an uncountable number of discontinuities, then what can happen?
Thank you for your help.