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Proof showing that if F is an antiderivative of f, then f must be continuous.

  1. Oct 26, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c).

    2. The attempt at a solution
    I attempted a proof by contradiction where I said that for the sake of contradiction we should let f have a discontinuity at point c, and I would like to use this to prove that F'(c) doesn't exist, but I'm not quite sure how a discontinuity on f affects F given that f is the derivative of F.
     
  2. jcsd
  3. Oct 26, 2014 #2

    LCKurtz

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    I haven't worked your problem out, but since you don't see how a discontinuity in ##f## affects ##F##, I would suggest you try some examples. For example look at what happens if ##f(x)=0,~0\le x\le \frac 1 2## and ##f(x) = 1,~\frac 1 2 < x \le 1##. What does ##F## look like in that case? That might give you some ideas.
     
  4. Oct 27, 2014 #3
    If I'm interpreting the problem correctly, it is basically saying the following:

    Assume ##F## is continuous on ##[a,b]## and differentiable on ##(a,b)## with ##f=F'##. Show that ##f## has no jump or removable discontinuities on ##(a,b)##.

    The "jump or removable" part of the problem is essential. Derivatives can be discontinuous; look at the function $$f(x)=
    \begin{cases}
    x^2\sin\frac{1}{x} & x\neq 0\\ 0 & x=0\end{cases}$$
    The function is differentiable everywhere, but the derivative is not continuous at ##0##.

    I would focus on one-sided limits of the derivative if I were you; i.e. show that if ##c\in (a,b)## and ##\lim\limits_{x\rightarrow c^\pm}f'(x)## exists, then ##\lim\limits_{x\rightarrow c^\pm}f'(x)=f'(c)##. That's the essence of what distinguishes the jump/removable discontinuities from the less "tame" kinds; the (one-sided) limit exists, but isn't equal to the value of the function.
     
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