I'm having trouble finding a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$
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1. $(t,x)\mapsto \partial_x^2 u(t,x)$ is $C^{0,\alpha}$;
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2. $(t,x)\mapsto \partial_x u(t,x)$ is not $C^{0,\alpha}$.
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3. $t\mapsto u(t,x)$ is $C^{1,\alpha}$.
All the statements must be true in a neighbourhood of $(0,0)$. I'm beginning to think that such functions do not exist so the next question. If $u$ has a partial derivative $C^{0,\alpha}$, is $u\in C^{0,\alpha}$?
Thanks.