A function not continuous but sill has antiderivative (necessary and sufficient condition for having antiderivative)We already know:1) if f(x) continuous on domain D then it is integrable and has antiderivative.2) f(x) is almost continuous (that is, the set of discontinuities has measure zero) is equivalent to integrability, but "almost continuous" doesn't guarantee having antiderivative.Now I am wondering is it possible for f(x) not continuous but still have antiderivative. Could you give me an example. It seems that people don't care about a function that have antiderivative, just integrable functions. Thank you.

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A function not continuous but sill has antiderivative (necessary and sufficient condition for having antiderivative)We already know:1) if f(x) continuous on domain D then it is integrable and has antiderivative.2) f(x) is almost continuous (that is, the set of discontinuities has measure zero) is equivalent to integrability, but &quot;almost continuous&quot; doesn&#039;t guarantee having antiderivative.Now I am wondering is it possible for f(x) not continuous but still have antiderivative. Could you give me an example. It seems that people don&#039;t care about a function that have antiderivative, just integrable functions. Thank you.

Christian Hawkins · Accepted Answer

Step 1Let f(x) be the derivative of the function F(x) such that   F  (  x  )  =      x    2    sin  ⁡  (  1      /    x  ) when   x  ≠  0, and   F  (  0  )  =  0. Then f(x) is not continuous at 0.Note that from the definition of the derivative, we can readily prove that F′(0) exists and is equal to 0. So   f  (  0  )  =  0.Step 2But for   x  ≠  0,   f  (  x  )  =      F    ′    (  x  )  =  (      x    2    )  (  −  1      /        x    2    )  cos  ⁡  (  1      /    x  )  +  2  x  sin  ⁡  (  1      /    x  )  =  −  cos  ⁡  (  1      /    x  )  +  2  x  sin  ⁡  (  1      /    x  )  ..The 2x sin(1/x) part approaches 0, but the cos(1/x) part wiggles wildly near 0. So f(x) is not continuous at 0.

A function not continuous but sill has antiderivative (necessary and sufficient condition for having

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