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