Assume that the function $f(t,x) \in C^{3,4}(\Omega)$, i.e. three times continuously differentiable as a function of $t$ and four times continuously differentiable as a function of $x$. Why does the partial derivative $f_{xxt}$ exist?
Background:
I am reading a proof which uses
- the Taylor expansion of $f(t,x)$ w.r.t. $t$ up to second partial derivative $f_{tt}$.
- the Taylor expansion of $f(t,x)$ w.r.t. $x$ up to the third partial derivative $f_{xxx}$.
- the Taylor expansion of $f_{xx}(t,x)$ w.r.t. $t$ up to the first partial derivative $f_{xxt}$.
So how does one actually make use of of the fact that $f \in C^{3,4}(\Omega)$?