Does this non-rigorous Pen-and-Paper explanation for Differentiability make sense? We can describe Continuity non-rigorously as not having to lift your pencil to draw a function.  Could we similarly describe Differentiability as not having to stop the pencil from moving to draw the function?  For example, when you draw $x^2$ you could draw it without ever having a velocity of 0 in your hands movement, but this is not possible with $|x|$, where you have to stop your motion at 0 to change course.
A couple posters have raised issues with this definition.. is there a way to resurrect it with better criteria? Is there a property that non-differentiable functions have when it comes to drawing them that could produce a stronger pen-and-paper rule?

I wouldn't describe it like that. Picture for example the function $f$ defined as
$$f(x)=\begin{cases} x& x<0\\
2x&x\geq0 \end{cases}$$
Here you can see that the derivative doesn't exist at zero but you don't need to invert the direction of the velocity of your hand when drawing it out.
Visually, I think that a differentiable function is "regular" in the sense that if you are drawing it you don't have to make "sudden" movements, like in the function $f$ above and the absolute value of x where you need to abruptly alter the angles involved in drawing the function
