Carathéodory’s Defi nition of the Derivative and Proof of the Chain Rule
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, F(0) = f’(x).
Show that Carathéodory’s definition of the derivative is equivalent to the definition on page 101.
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