Suppose that $f : \Bbb R \to \Bbb R$ is an integrable function. Show that the map $h(x)= \int_{- \infty}^x f(t) \ dt$ is continuous.

Asked Oct 20 '21 at 18:42

Active Oct 20 '21 at 18:42

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Suppose that $f : \Bbb R \to \Bbb R$ is an integrable function. Show that the map $h(x)= \int_{- \infty}^x f(t) \ dt$ is continuous.

What I’ve tried to do is show that $h$ is continuous with preimages, but that doesn’t seem like it’s going to work. For some $O$ open in $\Bbb R$ the preimage of $O$ under $h$ isn’t something that’s easy to work with. Is there some ”trick” that I’m not familiar with to prove continuity of this kind of map?

asked Oct 20 '21 at 18:42

Salmiac

3

Is it Riemann integrable, or Lebesgue integrable? – MH.Lee Oct 20 '21 at 18:45
@Nightflight It's not clear from the question, but there is the lebesgue-integral tag. – WhatsUp Oct 20 '21 at 18:47
@Nightflight It is Lebesgue integrable. – Salmiac Oct 20 '21 at 19:09

Suppose that $f : \Bbb R \to \Bbb R$ is an integrable function. Show that the map $h(x)= \int_{- \infty}^x f(t) \ dt$ is continuous.

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