Difference between revisions of "The Road to Reality Study Notes"

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

Penrose notes that Euler would have required $$C^\infty$$-smooth functions to be defined as functions, and then gives the function:<math>h(x)={\overset{0 if x\le0}{e^{-\frac{1}{x}} if x>0}</math> as an example of a $$C^\infty$$-smooth function but one that Euler would still not be happy with since it is two functions stuck together.