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(a) Show that a function h: I→ R is differentiable at a I if and only if there exists a function 1: I→ R which is continuous at x = a and satisfies h(x) - h(a) = (x)(x − a) for all x € I. (b) Use this criterion for differentiability (in both directions) to prove the Chain Rule.

(a) Show that a function h: I→ R is differentiable at a I if and only if there exists a function 1: I→ R which is continuous at x = a and satisfies h(x) - h(a) = (x)(x − a) for all x € I. (b) Use this criterion for differentiability (in both directions) to prove the Chain Rule.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Show that a function h: I→ R is differentiable at a I if and only if there exists a function 1: I→ R which is continuous at x = a and satisfies h(x) - h(a) = (x)(x − a) for all x € I. (b) Use this criterion for differentiability (in both directions) to prove the Chain Rule.
Transcribed Image Text:(a) Show that a function h: I→ R is differentiable at a I if and only if there exists a function 1: I→ R which is continuous at x = a and satisfies h(x) - h(a) = (x)(x − a) for all x € I. (b) Use this criterion for differentiability (in both directions) to prove the Chain Rule.
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