Is product of two Darboux functions Darboux?

As Dave Renfro points out there was a good deal of activity on the subject of Darboux functions in the past. Jack Ceder and Andy Bruckner at UCSB investigated such questions and produced very readable surveys on the subject. The literature is quite large.

The earliest answer to your question might be this that I have lifted from a Math Review of a paper:

"It is well known that the sum (and the product) of a continuous function and a Darboux function need not be Darboux in general [Th. Radakovič, Monatsh. Math. 38 (1931), 117–122; Zbl 1, 329]."

T. Radakovič, Über Darbouxsche und stetige Funktionen, Monatsh. Math. Phys. 38 (1931), 117–122.

The paper related to the quote is the article below. If you call such a Darboux function "bad" then you will appreciate their investigation of a "universally bad" one.

Kirchheim, Bernd; Natkaniec, Tomasz. On universally bad Darboux functions. Real Anal. Exchange 16 (1990/91), no. 2, 481–486.