A differentiable function whose derivative is not integrable

[jonsteadinho]

Homework Statement

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

Homework Equations

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.

 

[Dick]

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

 

[quasar987]

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?

 

[Dick]

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

 

[jonsteadinho]

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?

 

[Dick]

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.