Caratheodory Definition and Derivative Definition Problem The following is a sli

Question

Caratheodory Definition and Derivative Definition Problem

The following is a slight modification 01 the definition of the derivative of a function due to Caratheodory. DEFINITION Let f : D rightarrow R with x0 an clement of D that is also an accumulation point of D. Then f is differentiable at x0 iff there is a function k : D rightarrow R such that k is continuous at xo and for all x D, f(x) - f(x0) = k(x)(x - x0). Prove that the preceding definition is equivalent to the one given in this chapter. Use the definition of Caratheodory to prove the chain rule.

Explanation / Answer

I have written the answer on a paper and uploaded it as 6 pictures at the following links:

Question-1: PArt-1: http://i.imgur.com/Qu7sAhr.jpg

Part-2: http://i.imgur.com/qeCPdxl.jpg

Question-2: Part-1: http://i.imgur.com/SYUH8zy.jpg

Part-2: http://i.imgur.com/FTcyB3w.jpg

Part-3: http://i.imgur.com/yhfSht6.jpg

PArt-4: http://i.imgur.com/e29mEW7.jpg

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