- #1

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## Homework Statement

a) If ##f: [0,1] \rightarrow \mathbb{R}## is continous and ##\int^{b}_{a} f(x)dx = 0## for every interval ##[a,b] \subset [0,1]##, then ##f(x)=0 \forall x \in [0,1]##

b) Let ##f: [0,\infty) \rightarrow [0,\infty)## be continous. If ##\int^{\infty}_{0} f(x)dx## converges, then ##f(x) \rightarrow 0## when ##x \rightarrow \infty ##

## Homework Equations

## The Attempt at a Solution

For the a) part, my guess is that it is true ( I know that guesses don't mean much). I've tried to come up with a counter example such as

## f(x)=

\left\{

\begin{array}{ll}

1 & \mbox{if } x \in Q \\

0 & \mbox{if } x \in I

\end{array}

\right.

##

, which will satisfy the latter conditions, but the ones that say that the function is continuous and that ##f: [0,1] \rightarrow \mathbb{R}## (which would prevent me to use symmetric intervals around zero, and hence use sin(x) for example) make me think that it's true. However I'm not sure how to prove it.

b) Like before, the ##f: [0,\infty) \rightarrow [0,\infty)## condition makes me think it is true, since I cannot use bounded functions like sin(x) which would give me a counter example (here's a similar problem, but without the interval and continous conditions: http://math.stackexchange.com/questions/538750/if-the-improper-integral-int-infty-a-fx-dx-converges-then-lim-x→∞fx ). Also I have in my notes, before Cauchy's integral criterion (which states, for using it, that the function must be a decreasing one, something which is not specified here), that if

##S_{n}= \sum_{k=0}^{N} A_{k}## converges, then ##A_{k} \rightarrow 0##, but I cannot find a proof of this.

Any hint would be much appreciated.