Proof involving integrals equal to 0.

**amm345** · February 23rd 2010, 07:17 PM

Let

f be a bounded integrable non-negative function continous on [a, b].
Prove:
integral from a to b f(x) dx = 0 implies f(x) = 0 for every x

I believe I understand why it is true, I'm just not sure how to write the formal proof.

**Drexel28** · February 23rd 2010, 07:30 PM

Originally Posted by amm345

Let

f be a bounded integrable non-negative function continous on [a, b].
Prove:
integral from a to b f(x) dx = 0 implies f(x) = 0 for every x

I believe I understand why it is true, I'm just not sure how to write the formal proof.

What have you tried? Here is a weird approach.

Suppose that $f\ne 0$ then there exists some $x\in[a,b]$ such $f(x)\in(0,\infty)$ and since this set is open there exists some open ball $B_{\varepsilon}(f(x))\subseteq(0,\infty)$ and by continuity there exists some $B_{\delta}(x)$ such that $f\left(B_{\delta}(x)\right)\subseteq B_{\varepsilon}(f(x))\subseteq(0,\infty)$ . In other words, there exists some open ball around the point where $f(x)\ne0$ which is entirely positive. Take the concentric closed ball $B_{\frac{\delta}{2}}$ . This is a closed and bounded interval and thus, by the Heine-Borel theorem, compact. Thus, since $f:B_{\frac{\delta}{2}}[x]\mapsto\mathbb{R}$ is continuous we know that $f$ assumes a minimum on $B_{\frac{\delta}{2}}[x]$ . In particular, $\inf_{x\in B_{\frac{\delta}{2}}[x]}f(x)=\varepsilon>0$ . Thus, $\int_a^b\text{ }f=\int_a^{x-\frac{\delta}{2}}\text{ }f+\int_{x-\frac{\delta}{2}}^{x+\frac{\delta}{2}}\text{ }f+\int_{x+\frac{\delta}{2}}^b \text{ }f\geqslant 0+\frac{\varepsilon \delta}{2}+0>0$ . This of course is a contradiction.

**amm345** · February 23rd 2010, 07:33 PM

That was sort of my original approach, but when I checked with my instructer he said we could not use that theorem and I can't think of anything else.

**Drexel28** · February 23rd 2010, 07:39 PM

Originally Posted by amm345

That was sort of my original approach, but when I checked with my instructer he said we could not use that theorem and I can't think of anything else.

Which theorem is that?

**amm345** · February 23rd 2010, 07:40 PM

Oh sorry, the Heine-Borel theorem.

**Drexel28** · February 23rd 2010, 07:42 PM

Originally Posted by amm345

Oh sorry, the Heine-Borel theorem.

Are you serious? What's the point of having tools if you can't use them! What if I proved it first?

**amm345** · February 23rd 2010, 07:44 PM

I have no idea.. I couldn't think of anything else it is kinda ridiculous!

**Drexel28** · February 23rd 2010, 07:48 PM

Originally Posted by amm345

I have no idea.. I couldn't think of anything else it is kinda ridiculous!

Ok. I can think of one or two other ways....what are you currently doing in class?

**amm345** · February 23rd 2010, 07:57 PM

We started integration last week and we've mostly just covered the riemann integral, basic principles, the fundamental theorem and continuity.

**Drexel28** · February 23rd 2010, 08:03 PM

Originally Posted by amm345

We started integration last week and we've mostly just covered the riemann integral, basic principles, the fundamental theorem and continuity.

Hmm...Ok. I see an angle. Make a modified argument of mine and use the FTC.

**Moo** · February 23rd 2010, 11:48 PM

Hello,

Why can't we just say that (without HB theorem) :
if there exists $m\in[a,b]$ such that $f(m)>0$ , then by definition of the continuity, $\forall \epsilon>0, \exists \delta>0,\forall x\in[a,b], (|x-m|<\delta)\Rightarrow (|f(x)-f(m)|<\epsilon)$

In particular, if we let $\epsilon=\frac{f(m)}{2}$ , $|f(x)-f(m)|<\epsilon \Leftrightarrow -\tfrac{f(m)}{2}<f(x)-f(m)<\tfrac{f(m)}{2} \Leftrightarrow \tfrac{f(m)}{2}<f(x)<f(m)$

So there exists $\delta>0$ such that $\forall x\in]m-\delta,m+\delta[$ , $f(x)>\tfrac{f(m)}{2}>0$

Hence $\int_a^b f(x) ~dx=\int_a^{m-\delta} \underbrace{f(x)}_{\geqslant 0} ~dx+\int_{m+\delta}^b \underbrace{f(x)}_{\geqslant 0} ~dx+\int_{m-\delta}^{m+\delta} \underbrace{f(x)}_{> \frac{f(m)}{2}} ~dx$

So $\int_a^b f(x) ~dx>0+0+2\delta \cdot\tfrac{f(m)}{2}=\delta f(m)>0$

which is a contradiction.

So m can't exist, and hence $f(x)=0 ~,~ \forall x\in[a,b]$

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