| Smooth Function |
Article Index for Smooth |
Website Links For Smooth |
Information AboutSmooth Function |
| CATEGORIES ABOUT SMOOTH FUNCTION | |
| differential calculus | |
| smooth functions | |
| differential structures | |
|
Most of this article will be about Real -valued functions of one real variable. A discussion of the multivariable case will be presented towards the end. is a smooth function with Compact Support .]] DIFFERENTIABILITY CLASSES Consider an Open Set on the Real Line and a function ''f'' defined on that set with real values. Let ''k'' be a non-negative Integer . The function ''f'' is said to be of class ''Ck'' if the derivatives ''f To put it differently, the class ''C0'' consists of all continuous functions. The class ''C1'' consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a ''C1'' function is exactly a function whose derivative exists and is of class ''C0''. In general, the classes ''Ck'' can be defined Recursively by declaring ''C0'' to be the set of all continuous functions and declaring ''Ck'' for any positive integer ''k'' to be the set of all differentiable functions whose derivative is in ''Ck-1''. In particular, ''Ck'' is contained in ''Ck-1'' for every ''k'', and there are examples to show that this containment is strict. ''C∞'' is the intersection of the sets ''Ck'' as ''k'' varies over the non-negative integers. ''Cω'' is strictly contained in ''C∞''; for an example of this, see Bump Function or also below. EXAMPLES The function : is continuous, but not differentiable, so it is of class ''C0'' but not of class ''C1''. The function : is differentiable, with derivative : Because cos(1/''x'') oscillates as ''x'' approaches zero, ''f'' ’(''x'') is not continuous at zero. Therefore, this function is differentiable but not of class ''C1''. The Exponential Function is analytic, so, of class ''Cω''. The Trigonometric Function s are also analytic wherever they are defined. The function | ||
| : <math>p {K, M} | \sup_{x\in K}f^{(m)}(x)</math> |
|
|