The solution:
particular_script || true
Example:
$ cat /tmp/1.sh
particular_script()
{
false
}
set -e
echo one
particular_script || true
echo two
particular_script
echo three
$ bash /tmp/1.sh
one
two
three will be never printed.
Also, I want to add that when pipefail is on,
it is enough for shell to think that the entire pipe has non-zero exit code
when one of commands in the pipe has non-zero exit code (with pipefail off it must the last one).
$ set -o pipefail
$ false | true ; echo $?
1
$ set +o pipefail
$ false | true ; echo $?
0