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Under what conditions on $f(x,t)$ can we say that $f_{xt}$ or $f_{tx}$ exist? Does continuity of $f_x$ and $f_t$ suffice? (Note that I'm not talking about equality of the mixed derivatives.)

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Simply, $f_x$ must be differentiable in $t$ and $f_t$ must be diferentiable in $x$. –  user29999 Aug 5 '12 at 17:45
By definition $f_x(x,t)$ should be differentiable as a function of $t$ for $f_{xt}$ to exist, and mutatis mutandi for $f_{tx}$. –  hardmath Aug 5 '12 at 17:46
So continuity is certainly not enough. For instance $|t|$ is $f_x$ for $f(x,t)=-xt, t \in (\infty,0), xt, t \in [0,\infty)$. –  Kevin Carlson Aug 5 '12 at 17:49
That's disappointing. I hoped there was something about $f_{xx}$ existence that could have helped. Thanks anyway. –  TagWoh Aug 5 '12 at 17:49
@TagWoh : Perhaps something stronger than $f_x$ differentiable with respect to $t$ could be possible, but it is definitely the least requirement since it's the definition. But you definitely need to assume more than something about $f_x$ because if you assume nothing in $t$, the function could be really nice when you only vary the $x$ parameter but completely barbarian when the $t$ parameter varies. –  Patrick Da Silva Aug 5 '12 at 17:56
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