Let be a function, where and are Banach spaces. For , the function is said to be differentiable at if its derivative exists at that point. Differentiability at implies continuity at . If , then is said to be differentiable on if is differentiable at every point .
For the most common example, a real function is differentiable if its derivative exists for every point in the region of interest. For another common case of a real function of variables (more formally ), it is not sufficient that the partial derivatives exist for to be differentiable. The derivative of must exist in the original sense at every point in the region of interest, where is treated as a Banach space under the usual Euclidean vector norm.
If the derivative of is continuous, then is said to be . If the th derivative of is continuous, then is said to be . By convention, if is only continuous but does not have a continuous derivative, then is said to be . Note the inclusion property . And if the -th derivative of is continuous for all , then is said to be . In other words is the intersection .
Differentiable functions are often referred to as smooth. If is , then is said to be -smooth. Most often a function is called smooth (without qualifiers) if is or , depending on the context.
|Date of creation||2013-03-22 12:39:10|
|Last modified on||2013-03-22 12:39:10|
|Last modified by||Koro (127)|