Suppose you have a function and you would like to see that is . One way to show this is to find a function such that and
The implicit function theorem will imply that is .
Deterministic and Stochastic Optimal Control, Wendell H. Fleming, Raymond W. Rishel, page 8. Take a cost function and consider the calculus of variation problem of minimizing
over piecewise functions with the given fixed end points and . A piecewise function is called an extremal of (1) if it satisfies
for where is a constant and where denotes the variable of for which is substituted in (1) (the third variable).
Now let us assume that and consider an extremal of (1) that is also , i.e., is continuous. We will show using the implicit function theorem that must be , i.e., if the extremal is then it has to be as smooth as the cost function .
Now let us continue with our argument that must be as smooth as . The argument proceeds by induction. We already have the base case that is . Now let us suppose that it is for a . Then
is as well. By (2)
for some constant . Define
is and and is , it follows that is . We can rewrite (3) as and this is strictly positive by assumption. These imply that is at least as smooth as , thus is . It follows that is .