Editing The Road to Reality Study Notes (section)

=== 6.2 Slopes of functions ===

Differentiation is concerned with and calculates the rates that things change, or ‘slopes’ of these things.  For the curves given in section 6p1 above, two of the three do not have unique slopes at the origin and are said to be not ''differentiable'' at the origin, or not ''smooth'' there.  Further, the curve of theta(x) has a jump at the origin which is to say that it is discontinuous there, whereas <math>|x|</math> and <math>x^2</math> were continuous everywhere.

Taking differentiation a step further, Penrose shows us two plots which look very similar, but are represented by different functions, <math>x^3</math> and <math>x|x|</math>.  Each are differentiable and continuous, but the difference has to do with the curvature ([https://en.wikipedia.org/wiki/Second_derivative second derivative]) at the origin.  <math>x|x|</math> does not have a well-defined curvature here and is said to not be ''twice differentiable''.