Continue to a download a file after it has been interrupted with CURL:

curl -L -O -C - http://example.com/path/to/file.txt

From the man page:

-L, --location:

(HTTP) If the server reports that the requested page has moved to a different location (indicated with a Location: header and a 3XX response code), this option makes curl redo the request to the new place. If used together with --show-headers or --head, headers from all requested pages are shown.

-O, --remote-name:

Write output to a local file named like the remote file we get. (Only the file part of the remote file is used, the path is cut off.)

-C, --continue-at offset:

Resume a previous transfer from the given byte offset. The given offset is the exact number of bytes that are skipped, counting from the beginning of the source file before it is transferred to the destination. If used with uploads, the FTP server command SIZE is not used by curl.

Use "-C -" to instruct curl to automatically find out where/how to resume the transfer. It then uses the given output/input files to figure that out.