My impression is that $f$ is $C^k$ "close" to $g$ if $f$ is close to $g$, $f'$ is close to $g'$, ... $f^{(k)}$ is close to $g^{(k)}$.
However, my professor is being a bit nebulous on what "close" means. Would anyone be able to explain what it means for a function to be "close" to another? Does it imply some sort of conjugacy?
The context: My professor is attempting to define $C^k$ Structural Stability.