Example of a function in $L^{\infty}-L^p$

Question

This might be a silly question, but I have seen examples of functions in $L^p-L^{\infty}$ such as this, but I cannot come up with an example of a function $f$ such that $f\in L^{\infty}-L^p$.

So, my question is, does such a function even exist? If yes, then can you give an example? If no, then prove that no such $f$ exists. Note that you're free to use any measure space.

Answer 1

If the measure space is finite, then no such thing exists.

If the measure space is infinite, then any nonzero constant function will be in $L^\infty\setminus L^p$ for finite $p$