Example of a continuous function which is bounded and not contained in any $L_p$-space ($p\gt 0$)

Question

I'm struggling to find an example of a continuous function $f:(0,\infty)\to \mathbb R$ which is bounded, not contained in any $L_p$-space ($p\gt 0$) and goes to zero when x goes to infinity.

I need help.

Thanks a lot!

Answer 1

Consider $$ f(x)=\frac{1}{\log(x+3)} $$

Answer 2

Another example is $f:(1,\infty)\to\mathbb R$ defined by $$f(x)=\sum\limits_{n=1}^\infty2^{-n}x^{-1/n}.$$