Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

==== A “classical solution” means: ====
* the PDE holds pointwise (or in a strong sense),
* spatial derivatives exist,
* time derivative exists pointwise.

It does not automatically imply

t↦u(t)∈Xis continuous.t \mapsto u(t) \in X
\quad \text{is continuous.}t↦u(t)∈Xis continuous.
For example:
* a function may be smooth in space for each fixed ttt,
* yet jump in time in a strong norm.

So a “classical but XXX-discontinuous” object is logically possible.

Your definition of mild solution simply does not admit such objects.