I don't understand the definition of smooth function like a function that is derivable infinite times.
If "smooth" means that the graph has not sharp corners it is sufficient only the existence of the first derivative.
[Math] Why a function to be smooth requires more than one derivative
real-analysis
Best Answer
The word "smooth" is very over-used. It tends to mean "as differentiable as I need right now". In some contexts, it's used for $C^\infty$ (derivatives of all order exist and are continuous), in some for $C^1$ (first order derivative exists and is continuous). In some situation it could mean even more complicated things, like belonging to some Sobolev space.
Check the book you are using.