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Context is probably not important. What does it mean to say "$u_{a,b}$ depends $C^{m}$-ly on $a$ and $b$"? I know what it means to say a solution depends continuously on the data (change in solutions is less than or equal to change in data, informally).

Context: $u_{a,b} = u(at, x+bt)$ and $u(x,t)$ solves some PDE.

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    $C^m$ typically means that it is $m$ times continuously differentiable--so it has a continuous $m$th derivative--a stronger statement than merely continuous.2012-06-19
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    The partial derivatives of $u_{a,b}$ of order up to $m$ with respect to $a$ and/or $b$ exist and are continuous.2012-06-19

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