Let n ∈ℕ and consider the function fn: ℝ →ℝ defined by fn(x):={ x^nsin (1 / x) if x ≠ 0

step 1 Okay, so we know that f is an odd function and it is right continuous at x equals 0. Okay, so le

Let n ∈ℕ and consider the function fn: ℝ →ℝ defined by...

Let $n \in \mathbb{N}$ and consider the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $$ f_{n}(x):=\left\{\begin{array}{ll} x^{n} \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Prove the following: (i) If $n$ is odd and $k:=(n-1) / 2$, then $f_{n}^{(k)}$ exists and is continuous on $\mathbb{R}$, but $f_{n}^{(k+1)}$ does not exist at 0 . (ii) If $n$ is even and $k:=n / 2$, then $f_{n}^{(k)}$ exists on $\mathbb{R}$, but it is not continuous at $0 .$ (Compare Exercise 12 of Chapter 4.)

Let $n \in \mathbb{N}$ and consider the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
f_{n}(x):=\left\{\begin{array}{ll}
x^{n} \sin (1 / x) & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$
Prove the following: (i) If $n$ is odd and $k:=(n-1) / 2$, then $f_{n}^{(k)}$ exists and is continuous on $\mathbb{R}$, but $f_{n}^{(k+1)}$ does not exist at 0 . (ii) If $n$ is even and $k:=n / 2$, then $f_{n}^{(k)}$ exists on $\mathbb{R}$, but it is not continuous at $0 .$ (Compare Exercise 12 of Chapter 4.)

Key Concepts

Parity Effects on Smoothness

The parity (odd or even nature) of an exponent in functions that involve terms like x^n can significantly impact the smoothness and differentiability properties. In many cases, an odd exponent results in different cancellation and symmetry properties compared to an even exponent. These differences affect how many derivatives exist and whether those derivatives are continuous, particularly at points where the function's behavior changes, such as at the origin in piecewise defined functions.

Oscillatory Behavior in Functions

Oscillatory functions, such as sine and cosine, can rapidly change values, making the analysis of their limits and derivatives non-trivial especially when combined with other factors. When these oscillatory functions are multiplied by functions that vanish at certain points (like a power of x), the oscillations can be dampened. The rate at which the damping occurs influences whether the overall function or its derivatives remain continuous or even exist at those points.

Piecewise Defined Functions

Piecewise defined functions are those that have different expressions over different parts of their domain. Analyzing their differentiability properties often requires special attention at the boundary points where the definition changes. Establishing differentiability or continuity at these junctions can involve taking limits from both sides and ensuring that the definitions and their derivatives match up.

Higher-Order Derivatives

Higher-order derivatives involve taking the derivative of a derivative repeatedly. For functions that are defined differently at a special point (such as 0), checking the existence and continuity of these derivatives requires careful analysis. This process often involves understanding how the combination of the function’s factors (like power functions and oscillatory functions) behaves under repeated differentiation, particularly at the point where the function definition changes.

Differentiability

This concept refers to whether a function has a derivative at a given point. In analysis, verifying the differentiability of a function, especially at a point where the function is defined in a piecewise manner or behaves unusually, requires the limit definition of the derivative. Showing that the k-th derivative exists means that one must be able to differentiate the function k times and obtain a function that is well?defined in a neighborhood of the point of interest.

Please give Ace some feedback