Monday, 26 June 2017

calculus - Must a Function be Bounded for the Antiderivative to Exist over a Given Interval?




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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