Answer whether the following statements are True or False.
There exists $f \in C([0,1], \mathbb{R})$ satisfying the following two conditions:
- $\displaystyle{}\int_{0}^{1} f(x) d x=1;$ and
- $\displaystyle{}\lim _{n \rightarrow \infty} \int_{0}^{1} f(x)^{n} d x=0$.