# Increasing functions not necessarily continuous are measurable

Let $$f:\mathbb{R}\rightarrow\mathbb{R}$$ be an increasing function (does not have to be continuous). Show that $$f$$ is measurable.

I'm having a problem proving this if the function is not continuous. Any help would be greatly appreciated.

$$\{x: f(x) \leq a\}$$ is an interval for any real number $$a$$. In fact, it is either an interval of the type $$(-\infty,b)$$ or an interval of the type $$(-\infty,b]$$. To see this all have to show is that if $$f(x) \leq a$$ then $$f(y) \leq a$$ for any $$y \leq x$$. Note: to show that $$f$$ is measurable it is enough to show that $$\{x: f(x) \leq a\}$$ is measurable for each $$a$$ because sets of the type $$(-\infty,a]$$ generate the Borel sigma algebra.