I'm a bit puzzled by some of the conditions on integrability, differentiability, and continuity. I am working on the below question(s), and would like to see if my intuition checks out. As this is a homework question, I'd prefer hints over straight answers.

Let $f$ be [Riemann] integrable on $[a,b]$, and $c,x \in (a,b)$. Define $$F(x) = \int_a^xf$$ Then give proofs or counterexamples to the following:

a) If $f$ is differentiable at $c$, then so is $F$

b) If $f$ is differentiable at $c$, then $F^{\prime}$ is continuous at $c$

c) If $f^{\prime}$ is continuous at $c$, then $F^{\prime}$ is differentiable at $c$

Ok here are my thoughts:

a) I think I can boil this one down to the Fundamental Theorem of Calculus. Since $F(x)$ is an antiderivative, it must be differentiable.

b) Once again, by FTC, we know that $F^{\prime} = f$. Since differentiability implies continuity, $F^{\prime}$ must be continuous$

c) Not sure on this one. If $f^{\prime}$ is continuous, does it necessarily have an antiderivative? It's tempting to say yes but I really do not know.

Sorry for the homework question. Any help would be hugely appreciated.

  • $\begingroup$ For c) some antiderivative exists because is stated that $f$ is integrable at $c$. Because $f$ is continuous at $c$ then $F'(c)=f(c)$. For the next argument Im not totally sure: if $f'$ is continuous at $c$ then $f$ is differentiable in a neighborhood of $c$, and then $F'=f$ in this neighborhood, hence $F'$ is differentiable at $c$. $\endgroup$ – Masacroso Dec 4 '16 at 15:40
  • $\begingroup$ I can't tell if you really know the FTC precisely. Why do you say in a),b) that $F'=f?$ $\endgroup$ – zhw. Dec 4 '16 at 18:33

Some hints: a) If $f$ is diffentiable at $c,$ then $f$ is continuous at $c.$ And at any point where $f$ is continuous, you can say something nice about $F.$

b) You write $F'=f.$ Certainly that need not hold everywhere in $[a,b].$ What do you mean by this? I would also note that if $F'$ is continuous at $c,$ then $F$ would need to be differentiable everywhere in a neighborhood of $c.$ Why should we expect that to happen with the given hypotheses?

c) If $f'$ is continuous at $c,$ then $f'$ exists everywhere in a neighborhood of $c.$ Therefore $f$ is continuous in this neighborhood. The FTC then tells us $F'=f$ in this neighborhood. This implies $F''=f'$ in this neighborhood.


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