Yes, the L^p spaces are not vector spaces of functions, but essentially equivalent classes of functions that give the same result in an Lebesgue integral. For these reason, common operations on functions, like evaluating at a point or taking a derivative are undefined.

If you care about these you need something more restrictive, for example to study differential equations you can work in Sobolev spaces, where the continuity requirement allows you to identify an equivalent class with a well-defined function.

Thanks for the clarification