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I'm doing the exercise 3b, chapter 14 in Spivak's Calculus. It asks to prove or disprove the following statement:

If $f$ is differentiable at $c$, then $F'$ is continuous at $c$.

where $F(x)=\int_a^x f(y)dy$

$f$ is continuous at $c$ since it's differentiable at $c$ (by Theorem 9-1 "If $f$ is differentiable at $a$ then $f$ is continuous at $a$, page 156).

By the Fundamental Theorem of Calculus, $F$ is differentiable at $c$ and $F'(c)=f(c)$. And since $F$ is differentiable at $c$, $F$ is continuous at $c$.

But in the solutions the author says that we must assume that $f$ is continuous in an interval around $c$, otherwise $F'$ might not be continuous. How can $F$ be differentiable and not continuous at $c$? It contradicts the Theorem 9-1.

symplectomorphic
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super.t
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  • $F$ can be differentiable, but $F'$ not continuous. – zkutch Mar 08 '21 at 17:09
  • You will have that $F' = f$ at every point of continuity of $f$, but you may not have $F' = f$ at discontinuity points. So if $f$ is continuous in a neighborhood of $c$, then $F'=f$ in a neighborhood of $c$ and $F'=f$ (locally) is continuous at $c$ since $f$ is differentiable at $c$. However, if $f$ is not continuous in a neighborhood of $c$, then we don't necessarily have $F' = f$ near $c$, so the differentiability of $f$ is irrelevant since it may be that $F' \neq f$ near $c$. – nullUser Mar 08 '21 at 17:19
  • @nullUser: you wrote “if $f$ is not continuous in a neighborhood of $c$, then we don’t necessarily have $F’=f$ near $c$.” But it’s subtle. Since $f$ must be Riemann integrable on $[a,b]$, its set of discontinuities must have measure zero. As a result, every interval $(c, c+\epsilon)$ — ditto the other side — has a point where $f$ is continuous (and thus where $F’=f$). So we do have $F’=f$ at a lot of points near $c$. Moreover, as the proof linked at the end of my answer shows, even at points where $F’$ exists but differs from $f$, it can be shown that $F’$ does not differ too much from $f(c)$. – symplectomorphic Mar 10 '21 at 06:51

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Partial answer

How can $F$ be differentiable and not continuous at $c$?

It can’t. But this isn’t what Spivak claims; you’re mixing up $F$ and $F’$.

Spivak claims that $F’$, not $F$, need not be continuous at $c$. And in the solution manual he gives an explicit counterexample: $f$ can be Riemann integrable on $[a,b]$ and differentiable at an interior point $c$ without $F’$ being continuous at $c$. But as I discuss here, this counterexample only works because of Spivak’s very stringent definition of a limit at a point, which requires the function to be defined on an interval around the point (except possibly at the point itself). In his counterexample, the reason $F’$ isn’t continuous at $c$ is that $F’$ isn’t even defined at a series of points that converges to $c$.

Because of this, I find Spivak’s example “morally” disappointing: it works only because of a technicality. It’s more a consequence of a bad definition than a statement about the regularity of the function $F’$.

Indeed there is a more general definition of the limit of a function at a point that doesn’t require the function to be defined on a deleted interval around the point, and according to that definition, Spivak’s counterexample no longer works: his $F’$ is indeed continuous at $c$ using that other definition. In other words, near $c$ his $F’$ is pretty well-behaved wherever it’s actually defined.

I don’t know yet whether the claim “$f$ differentiable at $c$ implies $F’$ continuous at $c$” is true using this more general definition of a limit.

EDIT

User zhw proves here that the claim becomes true if we use the more general definition of continuity.

symplectomorphic
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  • may I ask what solution manual are you using? In mine he only sketched the counterexample function and didn't give its full definition. – super.t Mar 09 '21 at 08:25
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    @super.t: I have the same solutions manual. It took me a couple of hours to figure out what he meant by that terrible drawing. The essential point is that he says $F’$ “does not even exist” for points near $c$. – symplectomorphic Mar 09 '21 at 14:19