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1. Exactly one of the following requests is impossible. Decide which it is, and provide examples for the other three. In each case, assume f,g: R→ R are functions. (a) Functions f and g which are not differentiable at zero but where fg is differentiable at zero. (b) A function f not differentiable at zero and a function g differentiable at zero where fg is differentiable at zero. (c) A function f not differentiable at zero and a function g differentiable at zero where f + g is differentiable at zero. (d) A function f differentiable at 0 but not differentiable at any other point.

1. Exactly one of the following requests is impossible. Decide which it is, and provide examples for the other three. In each case, assume f,g: R→ R are functions. (a) Functions f and g which are not differentiable at zero but where fg is differentiable at zero. (b) A function f not differentiable at zero and a function g differentiable at zero where fg is differentiable at zero. (c) A function f not differentiable at zero and a function g differentiable at zero where f + g is differentiable at zero. (d) A function f differentiable at 0 but not differentiable at any other point.

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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**Problem 1** Exactly one of the following requests is impossible. Decide which it is, and provide examples for the other three. In each case, assume \( f, g : \mathbb{R} \to \mathbb{R} \) are functions. (a) Functions \( f \) and \( g \) which are not differentiable at zero but where \( fg \) is differentiable at zero. (b) A function \( f \) not differentiable at zero and a function \( g \) differentiable at zero where \( fg \) is differentiable at zero. (c) A function \( f \) not differentiable at zero and a function \( g \) differentiable at zero where \( f + g \) is differentiable at zero. (d) A function \( f \) differentiable at 0 but not differentiable at any other point.
Transcribed Image Text:**Problem 1** Exactly one of the following requests is impossible. Decide which it is, and provide examples for the other three. In each case, assume \( f, g : \mathbb{R} \to \mathbb{R} \) are functions. (a) Functions \( f \) and \( g \) which are not differentiable at zero but where \( fg \) is differentiable at zero. (b) A function \( f \) not differentiable at zero and a function \( g \) differentiable at zero where \( fg \) is differentiable at zero. (c) A function \( f \) not differentiable at zero and a function \( g \) differentiable at zero where \( f + g \) is differentiable at zero. (d) A function \( f \) differentiable at 0 but not differentiable at any other point.
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