In my Calculus class, we were given the following definition of antiderivative:
Let f be a function defined on an interval.
An antiderivative of f is any function F such that F′=f.
The collection of all antiderivatives of f is denoted ∫f(x)dx.
My question is, don't we have to say that f should be bounded in the definition?
If not, then f is not integrable by definition, so we can't say anything about ∫f(x)dx, right?
I'm not sure if I'm thinking about this the right way.
Answer
The existence of an antiderivative and being integrable are distinct (although related) concepts.
Takef:[0,1]⟶Rx↦{x2sin(1x2) if x>00 if x=0.Then f is differentiable, but f′ is unbounded. But, in particular, f is an antiderivative of f′.
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