is it syntactically valid to define a function with in another function?

In C, you cannot do that.
In C++ you can indirectly, e.g.

int f()
{
   struct S {
      int g() { return 1; }
   } s;
   return s.g();
}

You may declare a function within a function (so that it can be used), but it's definition must be placed elsewhere, e.g.

int f()
{
   extern int g();
   return g();
}
...
int g() { return 1; }

Cheers
Andi