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Message: <blockquote data-quote="mike" data-source="post: 740135">I'm having trouble finding a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$<ol> <li data-xf-list-type="ol"> 1. $(t,x)\mapsto \partial_x^2 u(t,x)$ is $C^{0,\alpha}$;</li> </ol> <ol> <li data-xf-list-type="ol"> 2. $(t,x)\mapsto \partial_x u(t,x)$ is not $C^{0,\alpha}$.</li> </ol> <ol> <li data-xf-list-type="ol"> 3. $t\mapsto u(t,x)$ is $C^{1,\alpha}$.</li> </ol>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.</blockquote>