## Is a derivative always Riemann integrable?

V is differentiable everywhere. The derivative V ′ is bounded everywhere. The derivative is not Riemann-integrable.

**Are all derivatives differentiable?**

It means that a function is differentiable everywhere its derivative is defined. So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable.

### Can a function be differentiable but not integrable?

To remove some possible confusion in the given answers: for your first question the answer is just “it is not possible to find any differentiable function that is not integrable”.

**What functions have no derivatives?**

In the case of functions of one variable it is a function that does not have a finite derivative. For example, the function f(x)=|x| is not differentiable at x=0, though it is differentiable at that point from the left and from the right (i.e. it has finite left and right derivatives at that point).

## Is every bounded function Riemann integrable?

Every bounded function f : [a, b] → R having atmost a finite number of discontinuities is Riemann integrable. 2. Every monotonic function f : [a, b] → R is Riemann integrable. Thus, the set of all Riemann integrable functions is very large.

**Does differentiability imply continuity?**

If a function is differentiable then it’s also continuous. This property is very useful when working with functions, because if we know that a function is differentiable, we immediately know that it’s also continuous.

### What does not differentiable mean?

A function is not differentiable at a if its graph has a vertical tangent line at a. The tangent line to the curve becomes steeper as x approaches a until it becomes a vertical line. Since the slope of a vertical line is undefined, the function is not differentiable in this case.

**What are non integrable functions?**

A non integrable function is one where the definite integral can’t be assigned a value. For example the Dirichlet function isn’t integrable. You just can’t assign that integral a number.

## What makes not differentiable?

**What kind of functions are not differentiable?**

The four types of functions that are not differentiable are: 1) Corners 2) Cusps 3) Vertical tangents 4) Any discontinuities Page 3 Give me a function is that is continuous at a point but not differentiable at the point. A graph with a corner would do.

### Which function is not Riemann integrable?

An unbounded function is not Riemann integrable. In the following, “inte- grable” will mean “Riemann integrable, and “integral” will mean “Riemann inte- gral” unless stated explicitly otherwise. f(x) = { 1/x if 0 < x ≤ 1, 0 if x = 0. so the upper Riemann sums of f are not well-defined.

**Why Dirichlet function is not Riemann integrable?**

The Dirichlet function is not Riemann-integrable on any segment of R whereas it is bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure).

## Can a function be discontinuous but differentiable?

It is possible for a differentiable function to have discontinuous partial derivatives. An example of such a strange function is f(x,y)={(x2+y2)sin(1√x2+y2) if (x,y)≠(0,0)0 if (x,y)=(0,0).

**What is differentiable and non differentiable?**

We say that f(x) is differentiable at x = a if this limit exists. If this limit does not exist, we say that a is a point of non-differentiability for f(x). If f(x) is differentiable at every point in its domain, we say that f(x) is a differentiable function on its domain.

### Can a function be discontinuous and differentiable?

If a function is discontinuous, automatically, it’s not differentiable.

**Why is this function not integrable?**

## How do you find non integrable functions?

Two basic functions that are non integrable are y = 1/x for the interval [0, b] and y = 1/x2 for any interval containing 0. The function y = 1/x is not integrable over [0, b] because of the vertical asymptote at x = 0. This makes the area under the curve infinite.

**What is an example of a non differentiable function?**

A function is non-differentiable when there is a cusp or a corner point in its graph. For example consider the function f(x)=|x| , it has a cusp at x=0 hence it is not differentiable at x=0 .

### What is meant by not differentiable?

**Which derivative is not Riemann integrable?**

MR0425042 (54 #13000) Goffman, Casper A bounded derivative which is not Riemann integrable. Amer. Math. Monthly 84 (1977), no. 3, 205–206.

## Is Volterra’s bounded derivatives integrable?

In 1881, Volterra constructed a bounded derivative that was not Riemann integrable. The existence of such functions later convinced Lebesgue that a better integration process needed to be devised. F(x) = ( x2sin(1=x); if x 6= 0 ; 0; if x = 0.

**Is F’ integrable with just finitely many discontinuities?**

Yes, there are examples with f’ bounded (see Goffman paper) but then you cannot get by with just finitely many discontinuities in f’ If f’ is bounded and continuous except at finitely many points, then f’ is integrable.

### How do you prove a function and its derivative?

I ask for pedagogical reasons. Results in basic real analysis relating a function and its derivative can generally be proved via the mean value theorem or the fundamental theorem of calculus. Proofs via FTC are often simpler to come up with and explain: you just integrate the hypothesis to get the conclusion.