A function with 
continuous derivatives
is called a 
function. In order to specify a 
function on a domain 
, the notation 
is used. The most common 
space is 
, the space of continuous
functions, whereas 
is the space of continuously differentiable
functions. Cartan (1977, p. 327) writes humorously that "by 'differentiable,'
we mean of class 
,
with 
being as large as necessary."
Of course, any smooth function is 
, and when 
, then any 
function is 
. It is natural to think of a 
function as being a little bit rough, but the graph of a
function "looks" smooth.
Examples of 
functions are 
(for 
even) and 
,
which do not have a 
st
derivative at 0.
The notion of 
function may be restricted to those whose first 
derivatives are bounded functions.
The reason for this restriction is that the set of 
functions has a norm which makes
it a Banach space,
 |
