What does C^r mean for a function of several variables?

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C^r for functions of several variables indicates that all partial derivatives, both mixed and unmixed, of order r must be continuous. This contrasts with the definition for functions of one variable, which focuses solely on the continuity of the rth derivative. The discussion confirms that the definition holds true across multiple variables. Understanding this distinction is crucial for analyzing the smoothness of multivariable functions. Overall, C^r signifies a higher level of continuity for derivatives in multivariable calculus.
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Wiki and mathworld only talk about C^r for functions of one variables, saying C^r is the space of function with and rth derivative that is continuous. But for a function of several variables to be C^r means that all its partial derivatives (mixed and not mixed) of order r are continuous, correct?
 
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Yes. Tht is correct.
 
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