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A (classical) solution of the wave equation $$ u_{tt}-c^2u_{xx}=0,\qquad (x,t)\in\mathbb{R}\times\mathbb{R}^*_+, $$ is required to be of class $C^2$. Why?

I mean, why one imposes that all second partial derivatives, even $u_{xt}$ which does not appear in the PDE, to be continuous?!

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