Answer to Question #308268 in Real Analysis for Nikhil

Question

Show that the function f defined on [0,1] by f(x)= (-1)^(n-1) for 1/(n+1) < x/n ≤ 1/n where (n=1,2,3...) is integrable on [0,1]

Answer 1

Let 's use the Darboux criterion. The function has points of discontinuity $\{\frac{1}{n},n\in\mathbb{N}\}.$

But since the function is bounded ($|f(x)|\leq1$) and the discontinuity points have continuous parts in the neighborhood (In other words, they are not dense), the Darboux criterion will still be fulfilled at the discontinuity points. So, the function is integrable.