A 
function is a function with 
continuous derivatives.
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,
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