Let n ∈ℕ. Find a function f:[-1,1] →ℝ for which f^(n)(0) exists, but f^(n+1)(0) does not. (Hint:

step 1 Since F is continuous in R, so it is continuous for all N in R. So F, 0 equal limit n tends to i

Let n ∈ℕ. Find a function f:[-1,1] →ℝ for which f^(n)(0)...

Let $n \in \mathbb{N}$. Find a function $f:[-1,1] \rightarrow \mathbb{R}$ for which $f^{(n)}(0)$ exists, but $f^{(n+1)}(0)$ does not. (Hint: Begin with the absolute value function and use part (ii) of Proposition $6.20$ repeatedly.)

Key Concepts

Absolute Value Function as a Prototype in Smoothness Analysis

The absolute value function is a standard example used to illustrate non-smooth behavior, as it is continuous everywhere but not differentiable at zero. By using this function and making systematic adjustments, one can create examples with controlled differentiability; for instance, raising the absolute value to a power can yield a function that is differentiable up to a certain order at a point, making it a useful building block in the study of higher derivative existence.

Differentiability and Order of Derivatives

This concept refers to the ability of a function to be differentiated a certain number of times; that is, a function can possess its first derivative, second derivative, and so on up to some order. The existence or non-existence of these derivatives at specific points is central to understanding the smoothness of the function and has implications in approximation, Taylor series expansion, and the behavior of the function near those points.

Construction of Functions with Specific Smoothness Class

This involves designing functions that are exactly k-times differentiable (i.e., belong to the class C^k) at a given point, while not being differentiable of order k+1. Techniques for such constructions often involve starting with a basic function—such as the absolute value function—and modifying it appropriately (for example, by taking suitable powers or using piecewise definitions) to ensure the desired differentiability properties at the target point.

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