Let f be defined by f(x) =0, 0 ≤x ≤1, =1, 1<x ≤2 . Then f is Riemann

step 1 Okay, so now we're looking at this, and we have it labeled as a function, which it is a function

Let f be defined by f(x) =0, 0 ≤x ≤1, =1,...

Let $f$ be defined by $$ \begin{aligned} f(x) & =0, & & 0 \leq x \leq 1, \\ & =1, & & 1<x \leq 2 . \end{aligned} $$ Then $f$ is Riemann integrable on $[0,2]$, but it is not the derivative of any function. For a more dramatic example, consider the function in Example 22.3(h), which cannot be a derivative by Exercise 19.N.

Let $f$ be defined by
$$
\begin{aligned}
f(x) & =0, & & 0 \leq x \leq 1, \\
& =1, & & 1<x \leq 2 .
\end{aligned}
$$

Then $f$ is Riemann integrable on $[0,2]$, but it is not the derivative of any function. For a more dramatic example, consider the function in Example 22.3(h), which cannot be a derivative by Exercise 19.N.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Intermediate Value Property (Darboux's Theorem)

Darboux's Theorem states that every derivative possesses the intermediate value property, meaning that if a derivative takes two different values at two points, then it must take every value between them at some point in the interval. This property is crucial in differentiating functions from arbitrary integrable functions; a function that fails to have this property, such as one with a jump discontinuity, cannot be the derivative of any function.

Riemann Integrability

A function is Riemann integrable if the set of its discontinuities has measure zero, or equivalently, if the upper and lower Riemann sums can be made arbitrarily close by an appropriate choice of partition. This concept means that even when a function has some discontinuities, it can still be integrated over a closed interval as long as those discontinuities do not ‘accumulate’ too badly.

Derivative as a Function

Not every function, even if integrable, can be expressed as the derivative of another function. A derivative must arise from a function via the process of differentiation, which implicitly imposes certain structural properties on the derivative, such as obeying the mean value theorem and Darboux's property, constraints that arbitrary functions may not satisfy.

Transcript

Please give Ace some feedback