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C^k Function


A function with k continuous derivatives is called a C^k function. In order to specify a C^k function on a domain X, the notation C^k(X) is used. The most common C^k space is C^0, the space of continuous functions, whereas C^1 is the space of continuously differentiable functions. Cartan (1977, p. 327) writes humorously that "by 'differentiable,' we mean of class C^k, with k being as large as necessary."

Of course, any smooth function is C^k, and when l>k, then any C^l function is C^k. It is natural to think of a C^k function as being a little bit rough, but the graph of a C^3 function "looks" smooth.

A C-k function

Examples of C^k functions are |x|^(k+1) (for k even) and x^(k+1)sin(1/x), which do not have a (k+1)st derivative at 0.

The notion of C^k function may be restricted to those whose first k derivatives are bounded functions. The reason for this restriction is that the set of C^k functions has a norm which makes it a Banach space,

 ||f||_(C^k(X))=sum_(n=0)^ksup_(x in X)|f^((n))(x)|.

See also

Banach Space, C-infty Function, Calculus, Continuously Differentiable Function, Continuous Function, Differential Equation

This entry contributed by Todd Rowland

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References

Cartan, H. Cours de calcul différentiel, nouv. éd., refondue et corr. Paris: Hermann, 1977.

Cite this as:

Rowland, Todd. "C^k Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/C-kFunction.html

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