Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$
Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$
Is it possible for $$\frac{d}{dt}f_{x_i},$$ to exist for all $i$ (as a classical time derivative), but for $f$ to not be continuous wrt. $t$ or classically differentiable wrt. $t$?
Here $f_{x_i} \neq 0$ for all $i$ (by this I am meaning that $f$ has spatial components, and is not just a function of time). Thanks.