Incremental reading of files that get rotated. If files are being written to it ignores the bytes written since the first `stat` call, avoiding double reads. Fairly untested so far, not even sure if it recognizes that the log rotation was performed. Mileage may vary, feel free to submit fixes.

logtail.sh

#!/bin/bash
function usage {
        echo Usage: $0 logfile.log logfile.log.1 [name]
        exit
}

if [ -z $1 ]; then
        usage
fi
if [ -z $2 ]; then
        usage
fi

LOGNAME=".offset"
if [ ! -z "$3" ]; then
        LOGNAME=".$3"
fi

LOG_SIZE=`stat -c %s $1`
LOG_OFFSET=0
if [ -f "$1$LOGNAME" ]; then
        LOG_OFFSET=`cat $1$LOGNAME`
else
        # first run, skip all data so far
        echo $LOG_SIZE > $1$LOGNAME
        exit
fi

if (( $LOG_SIZE > $LOG_OFFSET )); then
        CHANGED=`expr 1 + $LOG_SIZE - $LOG_OFFSET`
        tail -c +$LOG_OFFSET $1 | head -c $CHANGED
else
        LOG_SIZE2=`stat -c %s $2`
        if (( $LOG_SIZE2 > $LOG_OFFSET )); then
                tail -c +$LOG_OFFSET $2
        fi
        cat $1 | head -c $LOG_SIZE
fi

echo $LOG_SIZE > $1$LOGNAME