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Suppose $f$ is differentiable on $\mathbb{R}$ and its derivative $f'$ is continuous on the interval $[a,b]$. What constraints on $f$ would such condition give us?

mathchai
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  • Did you mean to ask "what constraints on $f'$ (the derivative of $f$) would follow? Because your title seems to be about properties of the derivative. – coffeemath Dec 14 '15 at 00:00
  • Sorry for the confusion. I'm wondering what the continuity of the derivative would tell us about the function $f$. It seems hard to me to relate this property back to $f$... – mathchai Dec 14 '15 at 00:06
  • I suggest changing the title to something like: What does the continuity of $f'$ tell us about $f$? [the first phrase "a function which is a derivative of another function" seems vague, in referring to two functions without giving either a name. – coffeemath Dec 14 '15 at 00:10
  • Perhaps a better way to pose this would look like this: Suppose that $F:[a,b]\to\mathbb R$. Find necessary and sufficient conditions on $F$ so that one may write $$F(x) = \int_a^x f(t),dt + C \ \ (a\leq x\leq b)$$ for some constant $C$ and some function $f$ belonging to [specified class]. Your problem is the version [specified class] = [continuous functions]. We could (and others have) ask for [specified class] = [BV functions] or = [functions in $L_p[a,b]$] etc. The goal is to find an intrinsic characterization. Obviously one doesn't want merely that $F'$ is continuous. – B. S. Thomson Dec 14 '15 at 00:13
  • I agree with your characterization. My question here is exactly what kind of instrinsic character would this function $F(x)$ have. – mathchai Dec 14 '15 at 00:47

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Here is an answer but I do not want it upvoted since it is not in the spirit of the question. What we really want is some property expressed more directly in terms of the values of $F$ that is both necessary and sufficient in order that $F'$ exists and is continuous. I offer this a "close but no cigar."

Theorem. A necessary and sufficient condition that $F:[a,b]\to \mathbb R$ is continuously differentiable on $[a,b]$ is that $F$ has a strong derivative at every point of that interval.

The strong derivative was introduced by Peano in 1892 (who called it a strict derivative and recommended that it should be used in elementary courses instead of the ordinary derivative). Unfortunately the "strong" terminology sticks, although "strict" would have been better. It is defined as $$D^\sharp F(x) = \lim_{(u,v)\to (x,x)} \frac{F(v)-F(u)}{v-u}$$ and sometimes called an "unstraddled" derivative since the limit here is not computed only for intervals $[u,v]$ that contain or straddle $x$, but for all "close" to $x$.

It has appeared before in a StackExchange question where Dave Renfro supplies a (probably) complete bibliography on the topic.

An equivalent answer, but sufficiently disguised that it may appear profound, is to express this in terms of the Dini derivatives.

Theorem. A necessary and sufficient condition that a continuous function $F:[a,b]\to \mathbb R$ is continuously differentiable on $[a,b]$ is that at every point of that interval one at least of the four Dini derivatives is continuous.

That is also not particularly intrinsic. I do have an intrinsic answer in mind but it is not going to make the OP happy so maybe this weak answer suffices.

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