I wrote this on an exam, is it correct?

[1MileCrash]

F(x) always exists and is differentiable as long as f(x) is continuous.

Do you agree?

[paulfr]

No, you have it wrong

Continuity and Differentiability
D => C but the converse C => D is not true

y = |x| is continuous at x=0 but not Differentiable there.


A function is Continuous ...
Informally..... if you can trace its graph without lifting your pencil
Formally .......if Limit f(x) as x-> a = f(a)

A function is Differentiable if
Informally..... if it can be approximated linearly (by a tangent line) at the point in question
Formally....... if the Limit of the definition exists
f ' (x) = Limit as dx -> 0 [ f(x+dx) - f(x) ] / dx

Existence and Differentiability are the same thing

[micromass]

No, you have it wrong

Continuity and Differentiability
D => C but the converse C => D is not true

y = |x| is continuous at x=0 but not Differentiable there.


A function is Continuous ...
Informally..... if you can trace its graph without lifting your pencil
Formally .......if Limit f(x) as x-> a = f(a)

A function is Differentiable if
Informally..... if it can be approximated linearly (by a tangent line) at the point in question
Formally....... if the Limit of the definition exists
f ' (x) = Limit as dx -> 0 [ f(x+dx) - f(x) ] / dx

Existence and Differentiability are the same thing

You misunderstood his question. He was talking about primitives and integrals.

1MileCrash: yes, a continuous function f:[a,b]\rightarrow \mathbb{R} always has a differentiable primitve function.

[shoescreen]

If by F(x) you mean an anti-derivative of f(x), you are correct.

[1MileCrash]

Alrighty, another question on the same subject but not from the exam.

Of course the nonelementary antiderivatives inspired that question. Is the lower limit in the integral sign of a nonelementary antiderivative kind of like the + C arbitrary constant for elementary antiderivatives?

[Mark44]

Alrighty, another question on the same subject but not from the exam.

Of course the nonelementary antiderivatives inspired that question. Is the lower limit in the integral sign of a nonelementary antiderivative kind of like the + C arbitrary constant for elementary antiderivatives?If I understand what you're asking, there's no connection between the lower limit of integration and the arbitrary constant.

What do you mean by "nonelementary antiderivative?" I get the feeling you're really asking about definite (w. limits of integration) versus indefinite (wo limits of integration) integrals.

[1MileCrash]

I can tell I wasn't clear, we're running on two different terminologies, what you're referring to, we call improper integrals. By elementary antiderivative, I mean an antiderivative that is just a normal polynomial/logarithmic/what have you function.

Now that I'm on an actual PC, I can show you.

The question was:

\int 8\sqrt{\frac{3}{4} sin^{2}\theta} d\theta

This function has no elementary antiderivative, but it does have an antiderivative. Show one, and explain why it has an antiderivative.

An antiderivative is:

\int^{\theta}_{0} \sqrt{\frac{3}{4} sin^{2}t} dt

Because that's a function of theta increasing at the rate of:
\ 8\sqrt{\frac{3}{4} sin^{2}\theta}

An my answer "why" was the opening of this thread.

So, my question is, since

\int^{\theta}_{0} \sqrt{\frac{3}{4} sin^{2}t} dt

Is increasing at the same rate and therefore has the same derivative whether I replace that 0 with 6, 22, 1000, is it like an arbirary constant for a normal antiderivative, since they are all antiderivatives of the original function no matter what the lower limit of the integral is?