Table of Contents
- 1 Can a function be integrable but not differentiable?
- 2 Does a function have to be differentiable to take the integral?
- 3 Where is FX not differentiable?
- 4 What is non integrable function?
- 5 Can you integrate any function?
- 6 What makes a function non differentiable?
- 7 How do you find whether a function is differentiable or not?
- 8 How do you know if a function is differentiable at x?
- 9 How do you prove that an integral is Riemann integrable?
- 10 What is the least upper bound of all integrals?
Can a function be integrable but not differentiable?
In practical terms, integrability hinges on continuity: If a function is continuous on a given interval, it’s integrable on that interval. This is just one of infinitely many examples of a function that’s integrable but not differentiable in the entire set of real numbers.
Does a function have to be differentiable to take the integral?
So, F(x) is an antiderivative of f(x). 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).
When can a function not be 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. Since the slope of a vertical line is undefined, the function is not differentiable in this case.
Where is FX not differentiable?
We can say that f is not differentiable for any value of x where a tangent cannot ‘exist’ or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative).
What is non integrable function?
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.
How many Antiderivatives can a function have?
Every continuous function has an antiderivative, and in fact has infinitely many antiderivatives. Two antiderivatives for the same function f(x) differ by a constant. To find all antiderivatives of f(x), find one anti-derivative and write “+ C” for the arbitrary constant.
Can you integrate any 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.
What makes a function non differentiable?
A function is non-differentiable where it has a “cusp” or a “corner point”. This occurs at a if f'(x) is defined for all x near a (all x in an open interval containing a ) except at a , but limx→a−f'(x)≠limx→a+f'(x) . (Either because they exist but are unequal or because one or both fail to exist.)
What functions are non differentiable?
Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. There are however stranger things. The function sin(1/x), for example is singular at x = 0 even though it always lies between -1 and 1.
How do you find whether a function is differentiable or not?
A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean? 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.
How do you know if a function is differentiable at x?
If f is Riemann integrable on the interval [ a, b], and if f is continuous at some x ∈ ( a, b), then F is differentiable at x, and F ′ ( x) = f ( x). Fundamental Theorem of Calculus–II. If a continuous function F: [ a, b] → R is differentiable at every x ∈ ( a, b) , and if its derivative F ′ is a Riemann integrable function, then
What does f(x)dx mean?
f(x)dx is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative).
How do you prove that an integral is Riemann integrable?
All the properties of the integral that are familiar from calculus can be proved. For example, if a function f: [ a, b] → R is Riemann integrable on the interval [ a, c] and also on the interval [ c, b], then it is integrable on the whole interval [ a, b] and one has ∫ a b f ( x) d x = ∫ a c f ( x) d x + ∫ c b f ( x) d x.
What is the least upper bound of all integrals?
Since L ( f, a, b) is the least upper bound of all such integrals, we must have L ( f, a, b) ≤ F ( b) − F ( a) . Let s ≤ f be a step function, and assume that a = x 0 < x 1 < ⋯ < x n = b are its partition points.