We know examples of functions (obviously we are in the context of real valued functions) which are continous but not derivable; the simplest is $x\mapsto|x|$. In particular we have a precise graphic idea of what "continous but not derivable in a given point" means. Let's informally call this $\mathcal C^0-\mathcal C^1$ graphic difference.
Can you provide some $\mathcal C^1-\mathcal C^2$ and $\mathcal C^2-\mathcal C^3$ graphic difference examples? Maybe someone knows a general rule to "see" $\mathcal C^n-\mathcal C^{n+1}$ graphic difference for real valued functions.
Many thanks!
If you look at a graph of $y=x^3$, to my eyes, as I read it left to right, I can see the downward concavity peter down to $0$ before turning into upward concavity.
But this is not what I see with $y=\frac{x^3}{|x|}=x|x|$. In that graph as I read it left to right, there is strong downward concavity until we reach $0$, and then the concavity jarringly switches to upward.
I'm sure my visual interpretation is at least partially biased from knowing the actual derivatives of these curves and having seen and thought about these examples lots of times over the years. But see what you think. $f$ with $f(x)=\frac{x^3}{|x|}=x|x|$ is $\mathcal{C}^1$ but not $\mathcal{C}^2$.