## Can differentiable function be integrable?

Ans: Not necessarily true. A function which is piecewise discontinuous is definitely integrable, but not the other way. Example: a step function is not differentiable but integrable. A Weierstrass function is integrable but not differentiable.

**Is the function differentiable on the interval?**

How To Determine Differentiability. By using limits and continuity! The definition of differentiability is expressed as follows: f is differentiable on an open interval (a,b) if lim h → 0 f ( c + h ) − f ( c ) h exists for every c in (a,b).

**Is the integral of a differentiable function differentiable?**

And, the theory of definite integrals guarantees that F(x) exists and is differentiable, as long as f is continuous. There is always an answer (there is always a function whose derivative is the function given to you, provided it is continuous).

### Can you integrate a function that is not differentiable?

For a function to be “differentiable” it must have a derivative at each point in its domain. As the integral of does not have a derivative at the integers, but the integers are part of its domain, the integral of is not differentiable on any interval that includes at least one integer.

**Can the Weierstrass function be integrated?**

The antiderivative of the Weierstrass function is fairly smooth, i.e. not too many sharp changes in slope. This just means that the Weierstrass function doesn’t rapidly change values (except in a few places). integrals, unlike derivatives, are highly insensitive to small changes in the function.

**Is derivative Riemann integrable?**

The derivative is not Riemann-integrable.

## Can a function be differentiable but not continuous?

There is no such function which is differentiable but not continuous. Because every differentiable function is continuous. So if a function is differentiable then it must be continuous.

**Where is a function not differentiable?**

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.

**Can you integrate every function?**

Not every function can be integrated. Some simple functions have anti-derivatives that cannot be expressed using the functions that we usually work with. One common example is ∫ex2dx.

### Can any function be integrated?

**Can a discontinuous function be integrated?**

Discontinuous functions can be integrable, although not all are. Specifically, for Riemann integration (our normal basic notion of integrals) a function must be bounded and defined everywhere on the range of integration and the set of discontinuities on that range must have Lebesgue measure zero.

**How do you prove a function is differentiable?**

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain. Let us look at some examples of polynomial and transcendental functions that are differentiable: f(x) = x4 – 3x + 5.

## What is a differentiable function that is not integrable?

Differentiable Function that is not Integrable : f ( x) = 1 x is differentiable in ( 0, 1), however it’s not integrable in this interval. That’s Undefined.

**Is the derivative of a function differentiable for a closed interval?**

A function f: A → R is differentiable for an accumulation point a ∈ A of A with derivative f ′ ( a), iff for each x n → a with x n ∈ A ∖ { a } you have f ′ ( a) = lim n → ∞ f ( x n) − f ( a) x n − a. So the answer is yes: You can define the derivative in a way, such that f ′ is also defined for the end points of a closed interval.

**Is the value of a piecewise function differentiable over an interval?**

However, since is piecewise defined, the limits on either side of use different definitions for the value of , and you have to make sure that the limits agree. A function is “differentiable” over an interval if that function is both continuous, and has only one output for every input.

### What is the definition of differentiability in math?

The definition of differentiability is expressed as follows: f is differentiable on an open interval (a,b) if lim h → 0 f ( c + h) − f ( c) h exists for every c in (a,b). f is differentiable, meaning f ′ ( c) exists, then f is continuous at c.