Editing The Road to Reality Study Notes (section)

=== 6.3 Higher derivatives; <math>C^\infty</math>-smooth functions ===
Looking closer at the concept of two derivatives of the same function (second derivative, or curvature), Penrose shows us the functions from 6p2 and their first and second derivatives.  Note that the first derivative of <math>f(x)</math>, written <math>f’(x)</math>, meets the x-axis at places of local minima or maxima and the second derivative of <math>f(x)</math>, written <math>f’’(x)</math>, meets the x-axis where the curvature goes to <math>0</math> and is said to be a point of inflection.

[[File:Fig 6p5.png|thumb|center]]

In general, a function can be smooth for many derivatives and the mathematical terminology for general smoothness is to say that <math>f(x)</math> is <math>C^n</math>-smooth.  It can be seen that <math>x|x|</math> is <math>C^1</math>-smooth but not <math>C^2</math>-smooth due to the discontinuity at the origin in the derivative. In general <math>x^n|x|</math> is <math>C^n</math>-smooth but not <math>C^{n+1}</math>-smooth.  In fact, a function can be <math>C^\infty</math>-smooth if it is smooth for every positive integer.  Note that negative integers for <math>x^n</math> immediately are not smooth for <math>x^{-1}</math> (discontinuous at the origin).  

Penrose notes that Euler would have required <math>C^\infty</math>-smooth functions to be defined as functions, and then gives the function:
:<math>h(x) = \begin{cases}
  0, & \mbox{if } x \le 0 \\
  e^{-\frac{1}{x}},  & \mbox{if } x > 0 
\end{cases}
</math>
as an example of a <math>C^\infty</math>-smooth function but one that Euler would still not be happy with since it is two functions stuck together.