Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous, and suppose that $\int_c^df=0$ for every interval $[c,d]\subset[a,b]$. Show that $f(x)=0$ for every $x\in[a,b]$.
I just need assistance writing a formal proof of this. I believe I have the general idea:
By the mean value theorem, there exists an $z\in[c,d]$ such that $f(z)(d-c)=0$. Because $\int_c^df=0$ for $\textit{every}$ subinterval $[c,d]\subset[a,b]$, it follows that $f(x)=0$ for every $x\in[a,b]$.
How do I make this rigorous?
You proved correctly that every interval $[c,d]\subset[a,b]$ (with more than one point) has a zero of $f$. Therefore, $\{x\in[a,b]\,|\,f(x)=0\}$ is dense in $[a,b]$. Since $f$ is countinuous, it follows from this that $f$ is the null function.