Riemann integrable function continuity points

Question

$f$ is Riemann integrable on $[a,b]$, $\int_a^bf(x)^2>0$. Prove that there is a point of continuity $x$ on $[a,b]$ that $f(x) \neq 0$. I provided the following: almost every point of $[a,b]$ is a continuity point. $F(x) = \int_a^bf(x)^2>0$ is differentiable in every continuity point and in such point $F'(x) = f^2(x)$. If $F(x) > 0$ then there is a point where $f^2(x) > 0$. It has a continuity point in some ball. Am I false everywhere?

Answer 1

One can show the following fact first:

If $f$ is Riemann integrable on $[a,b]$, and the set $\{x\in[a,b]:f(x)\neq0\}$ has measure $0$, i.e., $f$ is zero almost everywhere, then $\int_a^bf(x)\,dx=0.$

With this fact, if we assume $f(x)=0$ for any point of continuity $x$ of $f$, then by the fact that $f$ should be continuous a.e., it becomes a function that is zero a.e., and so is $f^2$, which contradicts the assumption that $\int_a^b (f(x))^2\,dx>0.$

Answer 2

We can use the following results:

Theorem 1: Let $f:[a, b] \to\mathbb{R} $ be Riemann integrable on $[a, b] $ with $\int_a^b f(x) \, dx>0$. Then there is a non-degenerate sub-interval $[c, d] $ of $[a, b] $ such that $f(x) >0$ for all $x\in[c, d] $.

Theorem 2: Let $f:[a, b] \to\mathbb{R} $ be Riemann integrable on $[a, b] $. Then there is a point $c\in[a, b] $ such that $f$ is continuous at $c$.

Use theorem 1 on $f^2$ to show that there is a sub-interval $[c, d] $ where $f^2>0$ and then $f$ is non-zero on that interval. Further by theorem 2 there is a point say $p\in[c, d] $ where $f$ is continuous at $p$ and we have $f(p) \neq 0$ because $f$ is non-zero on whole interval $[c, d]$.