Let $f:I\subset \mathbb R\to \mathbb R$ be a Riemann locally integrable function on the interval $I$. Show that, for a fixed $a\in I$ $$ x\mapsto F(x) =\int_a^x f(t)dt ~~~~ $$ is continuous on $I$. This a claim in Brezis book's page 205 lemma 8.2
Asked
Active
Viewed 52 times
1
-
Anything you tried already? – Thomas Jun 16 '17 at 19:07
-
Or this one: https://math.stackexchange.com/questions/429769/is-an-integral-always-continuous. – Martin R Jun 16 '17 at 19:10
-
@MartinR the result you sent is unclear and answer assume the function $f$ to be bounded. So is any Riemann integrable function bounded? If yes we are done – Guy Fsone Jun 16 '17 at 19:19
-
@Thomas I fail to use the epsilon delta definition – Guy Fsone Jun 16 '17 at 19:20
-
@GuyFabrice: Yes, a Riemann integrable function is bounded: https://math.stackexchange.com/questions/610054/if-a-function-fx-is-riemann-integrable-on-a-b-is-fx-bounded-on-a. – Note that it suffices to consider compact intervals $[a, b] \subset I$. – Martin R Jun 16 '17 at 19:21
-
I am wondering why that fact does not apply to the function $x\mapsto |x|^{-a}$ with $0<a<1$ which is Riemann integrable on $(-1,1)$ but bounded. – Guy Fsone Jun 17 '17 at 10:42