A 
function is a function
with 
continuous derivatives. In order to specify a 
function on a domain 
, the notation 
is used. The most common 
function space is 
, the function
space of continuous functions, whereas
is the function
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 
,
with the latter defined to be 0 at 
. These examples have 
continuous derivatives
but do not have a 
st
derivative at 0.
On an open interval 
, the 
functions whose derivatives
of orders 0 through 
are bounded form a Banach
space with norm
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