The following is an alternate definition of the derivative, due to Constantine Carathéod...

The following is an alternate definition of the derivative, due to Constantine Carathéodory.*

A function f is differentiable at x if there is a function F that is continuous at 0 and such that f(x+h) - f(x) = F(h) · h. In this case,

Show that Carathéodory’s definition of the derivative is equivalent to the definition on page 92.