I'm trying to find a smooth function $f: \mathbb{R} \times (0, \infty) \rightarrow \mathbb{R}$ which satisfies the following conditions:
1) $\lim_{t \rightarrow 0^+} t\cdot \left(\frac{\partial f}{\partial x}\right)^2$ exists and is not identically $0$ as a function of $x$.
2) $\lim_{t \rightarrow 0^+}t\cdot \left(\frac{\partial f}{\partial x}\right)\left(\frac{\partial f}{\partial t}\right)$ exists for all $x$.
3) $\lim_{t \rightarrow 0^+} t\cdot \left(\frac{\partial f}{\partial t}\right)^2$ exists for all $x$.
I've proved that $f$ can't be separable (i.e. $f(x,t)$ can't be of the form $f(x,t) = X(x)T(t)$). In fact I'm starting to believe that no such function can exist, but I haven't been able to prove it. Any help would be greatly appreciated!