How long a PID has been running (taking multiple inputs)
GNU long options are not POSIX, so instead of ps --pid "$1" --format etime=, write ps -p "$1" -o etime=.
kill -s 0 "$1" &gt;/dev/null 2&gt;&amp;1 is much faster than ps --pid "$1" &gt; /dev/null 2&gt;&amp;1, partly because kill is a builtin in most shells. Signal 0 is used to test for the existence of a process without sending it a signal.
Consider redirecting the echo messages to the appropriate output streams, i.e. stdout and stderr.
is_this_a_number, does_pid_exist: these are used directly in if clauses, so it'd read more naturally if they are renamed to is_a_number and pid_exists.
A minor point here, but I'd reverse the if-else statements to remove the ! negations.
Another minor point: during the short time span between testing the existence of a PID and querying the elapsed time since the process with that PID was started, the process with that PID could have finished running or even been substituted with a newly spawned one. You could avoid that by running the second command first, saving the output in a variable and then check for nullness, like thus:

echo "Process ID $arg_val"
running_time=$(how_long_pid_has_been_running "$arg_val" 2&gt;/dev/null | 
    trim_leading_and_trailing_spaces)
if [ -z "${running_time}" ]
then
    echo "This process does not exist."
else
    echo "${running_time}"            
fi

I've seen you write printf "\\n" in another post also, so I'm curious: is it any different from printf "\n"? And in any case, an echo would do the job just as well.