You don't need to do anything special to a currently running process at shutdown to continue downloading of the same file starting from the part of the file which was downloaded before. With HTTP downloads, you can skip part of it and start to download next block appending it to the file with already downloaded data.
You can find out how to use this simple technique if you look at my application HttpDownloader I provided for CodeProject with complete source code, please see how to download a file from internet
]. (This is C#. Too bad you did not indicate your language.)