It is not difficult to say the continuity of the simple function.
It is also not difficult to say the continuity of the geometry.
For example, we can say the following function is $C^0$ continuous.
$$f(x) = \begin{cases}x & \mbox{if }x \ge 0, \\ 0 &\text{if }x < 0\end{cases}.$$
We can say the corner in the following geometry is $C^0$ continuous
But how can we decide the continuity of discrete data?
For example, I think it should be $C^{-1}$ continuous.
But someone will say they can find a function that interpolates those point and guarantee the function to be $C^1$ continuous or even $C^\infty$ continuous
Then I have two questions:
Is there any definition that will tell: if the discrete data likes that, it is $C^{-1}$ continuous.
If I get the function that interpolates those points and is $C^1$ continuous or even $C^\infty$ continuous, I guess it will give lines which have very large oscillations. Based on the oscillations, can we say those data are not $C^1$ continuous or even $C^\infty$ continuous?
Thank you very much.