Counter-example for continuous function

Question

let $F:[a,b)\rightarrow \mathbb{R}$ be a continuous and non negative function. Is there exist an example of a $F(x)$ that $F^7(x)$ is not continuous at some point? If not, could you explain why? my intuition says it does not exist because multiplying continuous functions is still continuous

The question I asked was part of a bigger question im handling: given the same $F$ I defined above: assuming the integral $\int_a^b{e^{F(x)}}$ converge, does the integral: $\int_a^b{F(x)^7}$ converge? I'd like for a hint here because I'm kind of lost. my idea for this was that because $F(x)$ is non negative I could write: $\int_a^b{F(x)^7}=\int_a^b{e^{7ln(F(x))}}$ and then im missing a lemma why this could be proof

Answer 1

Hint for the second part: Since $F$ is nonnegative, $e^{F}\ge \dfrac{F^7}{7!}.$

Answer 2

As you wrote, the product of two continuous functions is continuous. You can deduce from this that, for each natural $n$, the product of $n$ continuous functions is continuous. In particular, the $7^{\text{th}}$ power of a continuous functions is continuous.