Why do we live on?
why not? because we live, we can hope.. and with hope, there can be
joy... for if a child who has been born in extreme poverty, in a
hut made up of refuse, where his family scrabbles through a
mountain of garbage for existence, can have the courage to smile,
to have the determination to hope for a better future.. to dare to
want to be happy.. we should strive to live on, and seek our own
happiness too 
If you live for somebody, what happens to the purpose of your
existence when that person dies or leaves?
we choose another purpose.. no matter how hard it is, we do.. and
really, we will.. because i've seen it with my own eyes when it
happened to a friend of the family twice.. and each time, she
picked herself up and went on..
herein lies one of the many paradoxical things in our lives.. we
can live for someone but that should not be the sole purpose of our
lives.. for in that, we could place an undeserved burden on the
other and on ourselves which we, may be happy enough to carry, but
may ultimately become a burden which is too much to bear...
for love should be liberating, i will walk the path we will take
because i choose to.. just like she walks that path because she
chooses to.. not because we are obliged to... and if one of us
leaves or dies, the other has a memory of where we have been, and
will continue the path chosen..
If you live for yourself, what exactly does living for yourself
mean?
as like the other answer, it should be tied in with living for self
and for others.. the selfish and narrow view would be to satisfy
one's own needs and wants.. however, that would be a shallow and
ultimately self-destructive way journey through life..
some have found that they find a sense of self-fulfillment and
self-satisfaction when they live a life of service to others.. in
that in service to others, they find a sense of peace and
satisfaction.. something which a pursuit to satisfy themselves only
led to a dead end..
maybe "living for yourself" in the selfish term, is a trap to
some.. especially those who still believe in goodness and
humanity.. but to others, "living for yourself" by seeking to do
good, they really do 
So... why do you live
on?
because i can.. because i want to.. because i believe that there
can be joy to be found, uncovered and created for both myself and
others.. where in a world, joy is in short supply, maybe in my
little way, i'm making this world a better place? i can't do much..
but if i can do some little good and make some people happy even if
only for a while, why not? 