Smoothness in one argument together with absolute continuity in the other argument imply joint continuity on a compact domain?

Question

Let $f(x,t) : [0,1] \times [0,1] \to \mathbb{R}$ be a function with the following properties:

1. For each $t \in [0,1]$ $f(\cdot,t)$ is periodic and $C^\infty$.
2. For each $x \in [0,1]$ and $n \geq 0$, $\partial_x^n f(x, \cdot)$'s are all absolutely continuous.

Then, I wonder if $f(x,t) : [0,1] \times [0,1] \to \mathbb{R}$ and all its partial derivatives with respect to $x$ are in fact jointly continuous in $(x,t) \in [0,1] \times [0,1]$.

I strongly suspect so, but it seems trickier to justify my guess since smoothness in $x$ depends on $t$..

Could anyone please help me?