Answer to Question #200207 in Real Analysis for Rajkumar

Question

Let f [: − 3,3 ] → R be defined by f (x)= 5x + x³ where [x] denotes the greatest integer ≤ x. Show that this function is integrable.

Answer 1

It is true. $f$ is continuous in $[-3,3]$ almost everywhere (it is not continuous in $\{3,4\}$, and $\{3,4\}$ no more than countable set). By the Lebesgue's criterion for Riemann integrability $f$ is integrable on $[-3,3]$.