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Let f : D = > R and c a cluster point of D. Then f is differentialble at c if an

ID: 1892799 • Letter: L

Question

Let f : D = > R and c a cluster point of D. Then f is differentialble at c if and only if there is a function K: D=> R such that k is continuous at c and for all x E D.

f(x) - f(c) = k(x) (x-c).

i) Prove that the Caratheododory definition is equivalent to the definition of the derivation given.
II) Use the Caratheodory definition to prove the chain rule.

Show detailed proof in answer.

Explanation / Answer

lim x->3 (f(x) - f(3))/(x - 3) = lim x->3 (1/x - 1/3)/(x - 3) = lim x->3 (3/(3x) - x/(3x))/(x - 3) = lim x->3 ((3 - x)/3x)/(x - 3) = lim x->3 -((x - 3)/3x)/(x - 3) = lim x->3 -1/3x = -1/9

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