Proving a function is identically zero

**Pinkk** · May 14th 2010, 03:03 PM

If $f$ is a continuous function on $[a,b]$ such that $\int_{a}^{b}fg = 0$ for all continuous functions $g$ . Then $f=0$ on $[a,b]$ .

So yeah, I know I have to invoke that $U(fg)=L(fg)=0$ but after that I'm stumped. Thanks.

**mabruka** · May 14th 2010, 03:09 PM

Here is an idea.

We have $\int_a^b f^2=0$ where $f^2\geq 0$ , so $f^2=0$ on $[a,b]$ .

Hence f=0 on [a,b]

What do you think?

**Pinkk** · May 14th 2010, 03:10 PM

I think that makes sense, yeah.

**Pinkk** · May 14th 2010, 03:43 PM

And I just want to make sure that proving there cannot be some point $x_{0}\in [a,b]$ where $f^{2} > 0$ . So suppose there is (at least) one point $x_{0}$ such that $f^{2}(x_{0}) > 0$ . Then since $f^{2}$ is continuous on $[a,b]$ , $f^{2} > 0$ on some interval $(x_{0} - \delta, x_{0} + \delta), \delta > 0$ , so $f^{2} > 0$ on the closed interval $[x_{0} -\frac{\delta}{2}, x_{0} + \frac{\delta}{2}]$ . Clearly $\inf \{f^{2}(x): x\in [x_{0} - \frac{\delta}{2}, x_{0} + \frac{\delta}{2}]\} > 0$ since the function must achieve a minimum at a point on the interval by its continuity on a bounded interval, so for any partition $P$ , $L(f^{2}, P) > 0$ , so $\int_{x_{0} - \frac{\delta}{2}}^{x_{0} + \frac{\delta}{2}} f^{2} > 0$ and since the function is zero everywhere else, then $\int_{a}^{b}f^{2} > 0$ , which is a contradiction.

I think this seems a little overboard but I just want to make sure if this is correct or if there is an easier proof that shows $f^{2}$ cannot be positive at any point. Thanks.

**Drexel28** · May 14th 2010, 03:48 PM

Originally Posted by Pinkk

And I just want to make sure that proving there cannot be some point $x_{0}\in [a,b]$ where $f^{2} > 0$ . So suppose there is (at least) one point $x_{0}$ such that $f^{2}(x_{0}) > 0$ . Then since $f^{2}$ is continuous on $[a,b]$ , $f^{2} > 0$ on some interval $(x_{0} - \delta, x_{0} + \delta), \delta > 0$ , so $f^{2} > 0$ on the closed interval $[x_{0} -\frac{\delta}{2}, x_{0} + \frac{\delta}{2}]$ . Clearly $\inf \{f^{2}(x): x\in [x_{0} - \frac{\delta}{2}, x_{0} + \frac{\delta}{2}]\} > 0$ since the function must achieve a minimum at a point on the interval, so for any partition $P$ , $L(f^{2}, P) > 0$ so then $\int_{x_{0} - \frac{\delta}{2}}^{x_{0} + \frac{\delta}{2}} f^{2} > 0$ and since the function is zero everywhere else, then $\int_{a}^{b}f^{2} > 0$ , which is a contradiction.

I think this seems a little overboard but I just want to make sure if this is correct or if there is an easier proof that shows $f^{2}$ cannot be positive at any point. Thanks.

That's similar to what I do. You can weaken this to something like if the above is true for any polynomial $g(x)$

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