Continuously differentiable function is integrable

[CStudent]

Hello everyone.

I'm studying to my final Calculus 2 exam.

One of the theorems I have to know is the one mentioned in the title:
Let f : [a, b] -> R s.t. f is continuously differentiable then f is integrable

Now, I look for some intuition for this theorem - why is a function integrable if it's continuously differentiable?
Why Differentiability itself isn't sufficient? And therefore what is special in being continuously differentiable for a function to be integrable?

Thanks in advance.

 

[pka]

I'm studying to my final Calculus 2 exam.
One of the theorems I have to know is the one mentioned in the title:
Let f : [a, b] -> R s.t. f is continuously differentiable then f is integrable
Now, I look for some intuition for this theorem - why is a function integrable if it's continuously differentiable?
Why Differentiability itself isn't sufficient? And therefore what is special in being continuously differentiable for a function to be integrable?

The fact that this in stated in the context of a Calculus II course is itself puzzling. The reason for that being one usually does not do a rigorous study of the theory of the integral such as by Saks or by Hildebrand in any undergraduate course.
Perhaps You should give more details about your course or at least name the textbook&author you are using.
Therefore, this answer is in the context of the usual & normal Calculus II course given in North America.
Here is the standard definition of continuously differentiable.
The following is a fairly statement in most calculus texts:
The Found Theorem of Calculus Suppose that \(\displaystyle f\) is a continuous function and
if \(\displaystyle \Phi (x) = \int_0^x {f(t)dt} \) then \(\displaystyle \Phi'(x)=f(x)\).

That is a Calculus II answer: The function \(\displaystyle \Phi(x)\) is the primitive function (or so called indefinite integral) of \(\displaystyle f\).
In advanced studies, textbooks texts mentioned above, the integral is defined as a number. It is a function of an interval.
If you know that I have missed your point, please give more detailed information.

 

[HallsofIvy]

Hello everyone.

I'm studying to my final Calculus 2 exam.

One of the theorems I have to know is the one mentioned in the title:
Let f : [a, b] -> R s.t. f is continuously differentiable then f is integrable

Now, I look for some intuition for this theorem - why is a function integrable if it's continuously differentiable?
Why Differentiability itself isn't sufficient? And therefore what is special in being continuously differentiable for a function to be integrable?

Thanks in advance.

Saying that a continuously differentiable function is integrable does NOT imply
that a function must be continuously differentiable. "If" is not the same as "only if".