Does Thomae's function have an antiderivative?Let f ( x ) = 0 if x is irrational and f ( x ) = 1 / q if x = p / q ∈ Q , p, q coprime natural numbers (restrict to [1,2] to make things easier). By boundedness and as it has at most countably many discontinuous points we know that f is integrable on [1,2]. Does it have antiderivative on such a compact?

Question

Does Thomae&#039;s function have an antiderivative?Let   f  (  x  )  =  0 if x is irrational and   f  (  x  )  =  1      /    q if   x  =  p      /    q  ∈      Q  , p, q coprime natural numbers (restrict to [1,2] to make things easier). By boundedness and as it has at most countably many discontinuous points we know that f is integrable on [1,2]. Does it have antiderivative on such a compact?

Patricia Curry · Accepted Answer

Step 1No; for the sake of contradiction suppose it did. Then, there is a function F such that       F    ′    =  f on the compact interval [1,2]. Note that derivatives have the intermediate-value-property, but we have       F    ′    (  1  )  =  f  (  1  )  =  1 and       F    ′        (                  3        2              )    =  f      (                  3        2              )    =            1      2      .Step 2This means that the entire interval   [      1    2    ,  1  ] must lie in the image of       F    ′    =  f. However, the image of f is a subset of the rational numbers. Thus, we have a contradiction.

Mara Cook · Accepted Answer

Step 1Here is another argument. Suppose there is an anti-derivative F. By Fundamental Theorem of Calculus we have   F  (  x  )  =  F  (  1  )  +      ∫          1              x        f  (  t  )    d  t for all   x  ∈  [  1  ,  2  ]. Since       F    ′    =  f it follows from above equation that we have       g    ′    =  f where g is defined by  g  (  x  )  =      ∫          1              x        f  (  t  )    d  tStep 2But notice that integral of Thomae function is 0 over any interval and hence g and g′ are both identically 0. And this gives our contradiction as   f  =      g    ′   is not identically zero (it is non-zero on rationals).

Does Thomae's function have an antiderivative? Let f ( x ) = 0 if x is irratio

Answered question

Answer & Explanation

New Questions in Calculus 1