Product of a derivative and a continuous function is a Darboux function Let $I\subset\mathbb{R}$ be an interval. Suppose that $f(x)$ has an antiderivative on $I$ and $g(x)$ is continuous on $I$, I have to prove that $f(x)g(x)$ is a Darboux function, having the intermediate value property.
($f(x)$ is said to have the intermediate value property on $I$, if for every $a,b\in I$, $a<b$, and $r\in [f(a),f(b)]$ (or $[f(b),f(a)]$), there exists $c\in [a,b]$ such that $f(c) = r$.)
If $f$ is nonzero everywhere, we actually know that $fg$ has an antiderivative (see here). The same applies if $f$ is bounded above of below (even just locally) since we can add a constant to $f$. But in general, $fg$ may not have an antiderivative. An example can be found here.
I have no idea what to do with this problem. Any help appreciated.
 A: Here is some background source for this problem.  Not a solution (alas) but you could use this material to find the neatest one.
First: it is true that the product of a derivative and a continuous function must be a Darboux function.  More precisely that product is a Darboux Baire 1 function.  As the OP notes the product of a derivative and a continuous function need not, however, be a derivative.  If it were then this would be obvious (it isn't).
The first person to ask such questions seems to have been Isaiah Maximoff in the 1930s.
He proved this:

If $f$ is a real-valued function on $R$ then $f$ is a Darboux function
of the first Baire class if and only if there is a homeomorphism $h$
of $R$ onto $R$ such that $f∘h$ is a derivative.

In a series of papers (some listed below) he gave several characterizations of the class of Baire 1 Darboux functions.  From those you can deduce this:
Theorem. The sum and the product of a Darboux function in Baire class 1 and a continuous function  are also Darboux functions in Baire class 1.
It can be deduced in a different way too (can't everything?).  Bruckner, Ceder and Weiss in reference [1] explored this problem:

Every real-valued function can be expressed as the pointwise limit of
a sequence of Darboux functions.  Characterize those functions which
can be expessed as the uniform limit of a sequence of Darboux
functions.

Their solution of this problem also can be applied here.  As they stated in the paper:

It follows from Theorem 4.4 and the corollary that the sum and
product of a Darboux function in Baire class 1 and a continuous
function  are also Darboux functions in Baire class 1.   This also
follows from a criterion of Maximoff ([2], p. 260).

Acknowledgement.  My thanks to Andy Bruckner for pointing to his 1966 paper with Jack Ceder and Max Weiss for background on this problem.
REFERENCES:
[1]  Bruckner, A. M.; Ceder, J. G.; Weiss, Max.
Uniform limits of Darboux functions.
Colloq. Math. 15 (1966), 65–77.
[2] Maximoff, Isaiah.  Sur les fonctions ayant la propriété de Darboux. Prace Matematyczno-Fizyczne 43 (1936), 241-265.
[3] Maximoff, Isaiah. On approximately continuous functions. Bull. Amer. Math. Soc. (1939), 264-268.
[4] Maximoff, Isaiah.
On functions of class 1 having the property of Darboux.
Amer. J. Math. 65 (1943), 161–170.

Postscript.  Maximoff has 18 publications listed from 1939-1944.  The only indications of where he was is  "TCHEBOLSSARY, TCHAPAEWSKIPOSELOK, U.S.S.R."  I can't find any biographical detail about him.  Those dates and his presence in the soviet union in those years suggests a sad story.
