Graphical explanation of the difference between $C^1$ and $C^2$ function? We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary explanations are given:

A continuous function is a function whose graph has no "holes" or "jumps", and a differentiable function is a function whose graph has no "corners".

This is a non continuous function:

This is a non differentiable continuous function: 

And this is a differentiable continuous function:

Is there a "graphical" or intuitive explanation of the difference between a $C^1$ function and a differentiable function with discontinuous derivatives? What about a function that is $C^1$ but not $C^2$ because it does not have second derivatives? Or what about a function that is $C^1$ and has second derivatives but they are not continuous? What about the difference between a $C^1$ function and a $C^{\infty}$ function? 

If $f$  is everywhere differentiable, but $f'$ is not continuous at some point, then $f'$ has to be very discontinuous there, because otherwise $f'$ could not satisfy the intermediate value requirement. As an example consider the function $f(x):=x^2\sin{1\over x}$ $(x\ne0)$, $f(0):=0$.
On the other hand there are beautiful functions which are $C^1$, but not everywhere twice differentiable, e.g., the function $f(x):=0$ for $x\leq0$ and $:=x^2$ for $x\geq0$. Here $f''(0)$ is undefined, and has a jump discontinuity there.
