Consider the function f: ℝ →ℝ defined by f(x):={ x^2sin(1 / x^2) if x ≠ 0 0 i

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Consider the function f: ℝ →ℝ defined by f(x):={ ...

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $$ f(x):=\left\{\begin{array}{ll} x^{2} \sin \left(1 / x^{2}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Show that $f$ is differentiable on $\mathbb{R}$, but for any $\delta>0, f^{\prime}$ is not bounded on $[-\delta, \delta] .$ Thus $f^{\prime}$ has an antiderivative on the interval $[-1,1]$, but it is not Riemann integrable on $[-1,1]$.

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
f(x):=\left\{\begin{array}{ll}
x^{2} \sin \left(1 / x^{2}\right) & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$
Show that $f$ is differentiable on $\mathbb{R}$, but for any $\delta>0, f^{\prime}$ is not bounded on $[-\delta, \delta] .$ Thus $f^{\prime}$ has an antiderivative on the interval $[-1,1]$, but it is not Riemann integrable on $[-1,1]$.

Key Concepts

Differentiability

Differentiability of a function involves the existence of a derivative at every point in its domain. This concept requires that the limit of the difference quotient exists as the interval approaches zero, ensuring the function has a linear approximation at each point. Even if the function exhibits complex or oscillatory behavior, the careful analysis of limits can often prove differentiability on the entire domain.

Unbounded Derivative

A function can be globally differentiable yet possess a derivative that is unbounded on certain intervals. This means that, around some points, the values of the derivative can become arbitrarily large in magnitude. The unboundedness of the derivative is a subtle phenomenon that highlights the difference between the existence of a derivative at every point and the uniform control over the derivative's size.

Antiderivative

An antiderivative of a function is another function whose derivative equals the original function. The existence of an antiderivative is guaranteed under various conditions, such as when a function is continuous. However, even if a function has an antiderivative, it does not necessarily imply that the function is well-behaved in terms of integration over an interval.

Riemann Integrability

Riemann integrability of a function over an interval typically requires the function to be bounded with a set of discontinuities of measure zero. A function with unbounded oscillations or discontinuities in every subinterval may fail to be Riemann integrable. This distinction is crucial, as a function related to the derivative of a differentiable function might not satisfy the conditions needed for Riemann integration.

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