From his website:
The Continuing End of Eternity was written whilst I was working partly in France and I found myself with spare time in hotels of an evening. I have always loved The End of Eternity and wanted to find some way of linking it into the rest of Asimov's Science Fiction Works. It was purely for my entertainment until someone commented, over a year later on one of the newsgroups, that The End of Eternity didn't fit. I posted my story and got some very pleasant responses. One person even liked it enough to translate it into French, which is appropriate considering where I wrote it! I have put a link to Jean-Claude's French Asimov site. I hope you get as much satisfaction reading it as I got writing it.