# 1D Finite element method: Function contineously differentiable?

I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin.

On page 50 and 51 which I attach as a screenshot below they show for an example functional, that minimizing the functional will lead to the solution of the pde in 3.12.

My question is: In the step 3.8 to 3.9 one did an integration by parts. For this step $$\alpha \frac{d\phi}{dx}$$ has to be differentiable? If I assume that $$\alpha$$ to be differentiable, then $$\phi$$ has to be differentiable twice.

When I minimize the functinal in practice I discretize space and assume in all discretized space intervals for example linear functions (linear element functions). With these linear functions I stick together my $$\phi$$. But this $$\phi$$ is then of course only contineous, but not twice differentiable!

Does anyone know a solution to this? Maybe it is okay if the discretization is fine enough?