Strictly Increasing function is measurable

Question

I am having trouble proving the following questions:

1. Prove that if $f$ is strictly increasing on $[a,b]$ then $f$ is measurable (do not assume that $f$ is continuous).
2. Using the above, prove that every nondecreasing function $g$ is measurable.

I know that I need to prove that the set $\{x \in [a,b] : f(x) > c\}$ (or some variation of it) is measurable.

Any hints would be most helpful!

Answer 1

See W. Rudin: Principles of math. analysis, p. 83, second edition: the set of points of discontinuities of a monotone function on (a,b) is most countable. Thus f is continuous almost every where and hence measurable.