A differentiable function whose derivative is not integrable

In summary, the question is whether a differentiable function g on [a,b] with f = g' can have a non-integrable function f. The conversation discusses possible examples of such a function, including a function that oscillates very fast near an endpoint but remains differentiable, and a function that is differentiable everywhere except at a single point. It is concluded that such a function can exist and be differentiable but not integrable.

Homework Statement

Suppose g is a differentiable on [a,b] and f = g', then does there exist a function f which is not integrable?

The Attempt at a Solution

I've tried to look at pathological functions such as irrational, rational piecewise functions. but the g would not be differentiable. Moreover, it seems intuitive that the derivative of a function is integrable. Please help.

How about a function that oscillates REALLY FAST as it approaches an endpoint but does it in such a way as to remain differentiable?

Dick said:
How about a function that oscillates REALLY FAST as it approaches an endpoint but does it in such a way as to remain differentiable?

Can such a function exist and be differentiable at the endpoint?

f(x)=x^2*sin(1/x^3) for x in (0,1], f(0)=0? What's wrong with that?

Thanks for the advice Dick. I thought of a function g(x) = x^2sin (1/x) at x not equals to 0 and g(x) = 0 when x = 0

Such a function is differentiable and g'(x) = f(x) = 2x sin (1/x) - cos (1/x) when x not equal 0 and 0 when x = 0

Yet, can't we say that such a function f is integrable on [-1,1]? since it is continuous at all points except at 0 and it is bounded on [-1,1] since both sin and cos functions are bounded. Thus, can't we choose a partition such that U(f,P)-L(f,P) < epsilon for any epsilon > 0?

Yes. I think so. If you look at my response to quasar987, that's why I picked one that oscillates even faster and is obviously unbounded and not integrable. I can make it oscillate even faster if you want.

1. What is a differentiable function?

A differentiable function is a function that is smooth and continuous, meaning that it has a well-defined derivative at every point in its domain.

2. What does it mean for a derivative to be integrable?

If a derivative is integrable, it means that the area under the curve of the derivative function can be calculated and has a finite value. In other words, the derivative function can be integrated to find the original function.

3. Can a differentiable function have a derivative that is not integrable?

Yes, it is possible for a differentiable function to have a derivative that is not integrable. This means that the derivative function cannot be integrated to find the original function.

4. Why is it important to have a differentiable function with an integrable derivative?

Having a differentiable function with an integrable derivative is important because it allows us to use the fundamental theorem of calculus, which states that integration and differentiation are inverse operations. This makes it easier to solve problems and calculate values of the original function.

5. What are some real-life examples of differentiable functions with non-integrable derivatives?

Some examples of differentiable functions with non-integrable derivatives include the absolute value function, the Dirichlet function, and the Weierstrass function. These functions are important in mathematics and physics, but are not easily integrable and require more advanced techniques to be evaluated.

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