The meanings of some symbols in "Calculus of variations"

Question

Could someone tell me the meanings of the "C" and its superscript "1" and subscript "0" in the equation which I have marked.

Thank you very much!!!

Answer 1

A function is in $C[a,b]$ (also written $C^{0}[a,b]$) if it is continuous on the interval $[a,b]$. A function is in $C^{n}[a,b]$ if it has $n$ derivatives, all continuous. The interval can be omitted if one means all of whatever set one is writing about. For example, $|x|$ is in $C^{0}$ because it is a continuous function, but $|x|\not\in C^{1}$ because its derivative is not continuous. Sometimes $C^{\infty}$ is written to denote infinitely smooth/differentiable functions. The $0$ subscript can mean different things in different texts, but I often see it mean that the functions in the set have boundary conditions equal to 0 on the first $n-1$ derivatives, including the original function.