Indeed, the function-space is comprised of piecewise third-order polynomials. So a third-order polynomial on each element. Within each element, you can take as many derivatives as you want, as a polynomial is infinitely differentiable. However, across element boundaries, the functions are “only” continuous. The first derivative of a function in your Q jumps from element to element. That is causing an integration inconsistency. The keywords to look for here are H1-sobolev space, C0 continuity, and FEM for higher-order PDEs. Most classical textbooks cover this material.
If I recall correctly, I touch upon that in this lecture:
