Is $t \mapsto \frac{\partial^2 f}{\partial x_1 \partial x_2}(x,t)$ continuous if $f \in C^{\infty}(\mathbb{R}^2 \times [0;+\infty[,\mathbb{R})$?

Question

Conditions: $f \in C^{\infty}(\mathbb{R}^2 \times [0;+\infty[,\mathbb{R})$.

Is the function $\displaystyle t \longrightarrow \frac{\partial^2 f}{\partial x_1 \partial x_2} (x_1,x_2,t)$ continuous on $[0;+\infty[$ for any $(x_1,x_2) \in \mathbb{R}^2$?

I don't find any counter-example, but I am not sure. And I have been looking for on the web for a while now. May someone have any demonstration?