Antiderivative of infinitely often differentiable functionDefine interval I = ( 0 , 1 ), C ∞ ( I ) as a ring of infinitely often differentiable functions from I to R , and C C ∞ ( I ) ⊂ C ∞ ( I ) an ideal of functions having compact support. Set d : C ∞ ( I ) → C ∞ ( I ) and d ¯ : C C ∞ ( I ) → C C ∞ ( I ) a usual differentiation.Is it true that Im ( d ) = C ∞ ( I ) and Im ( d ¯ ) = C C ∞ ( I )? In other words, is it true that antiderivative F of f ∈ C ∞ ( I ) also lies in C ∞ ( I ) and same for C C ∞ ( I )? If not, how can one describe Im ( d ) and Im ( d ¯ )?

Question

Antiderivative of infinitely often differentiable functionDefine interval   I  =  (  0  ,  1  ),       C          ∞        (  I  ) as a ring of infinitely often differentiable functions from I to       R  , and       C    C          ∞        (  I  )  ⊂      C          ∞        (  I  ) an ideal of functions having compact support. Set   d  :      C          ∞        (  I  )  →      C          ∞        (  I  ) and             d      ¯        :      C    C          ∞        (  I  )  →      C    C          ∞        (  I  ) a usual differentiation.Is it true that   Im  (  d  )  =      C          ∞        (  I  ) and   Im  (            d      ¯        )  =      C    C          ∞        (  I  )? In other words, is it true that antiderivative F of   f  ∈      C          ∞        (  I  ) also lies in       C          ∞        (  I  ) and same for       C    C          ∞        (  I  )? If not, how can one describe   Im  (  d  ) and   Im  (            d      ¯        )?

Quinn Everett · Accepted Answer

Step 1Assume that   f  ≥  0 and   f  ∈      C          ∞        (  0  ,  1  ) and let   F  ∈      C          0              ∞        (  0  ,  1  ) be its anti-derivative, assume further that             ∫              0                    1              f    (    x    )    d    x    &amp;gt;    0  , then for some large N, we have   F  (  x  )  =  0 for all   x  ∈  (  0  ,  1      /    N  )  ∪  (  1  −  1      /    N  ,  1  ). Now express   F  (  x  )  =            ∫              a                    x              f    (    t    )    d    t   for some   a  ∈  (  0  ,  1  ).Step 2And we know that             ∫              1                  /                n                    1        −        1                  /                n              f    (    t    )    d    t    →          ∫              0                    1              f    (    t    )    d    t   as   n  →  ∞, find some   M  &amp;gt;  0 large enough such that   1      /    M  ∈  (  0  ,  1      /    N  ) and   1      /    M  &amp;lt;  a, and that             ∫              1                  /                M                    1        −        1                  /                M              f    (    t    )    d    t    &amp;gt;    0  , then                               ∫                      1                          /                        M                                1            −            1                          /                        M                          f        (        t        )        d        t        =                  ∫                      1                          /                        M                                a                          f        (        t        )        d        t        +                  ∫                      a                                1            −            1                          /                        M                          f        (        t        )        d        t        =        −        F        (        1                  /                M        )        +        F        (        1        −        1                  /                M        )        =        0        ,            , a contradiction, so F cannot be compactly supported.

Sonia Gay · Accepted Answer

Explanation:The antiderivative of   f  ∈      C    ∞    (  I  ) is also in       C    ∞    (  I  ). It is not in       C    c    ∞    (  I  ) (even if   f  ∈      C    c    ∞    (  I  )) unless       ∫    I    f  =  0.

Antiderivative of infinitely often differentiable function Define interval I = ( 0

Answered question

Answer & Explanation

New Questions in Calculus 1