Rosemary Dies in Singapore

I've tried several times to grow Rosemary in Singapore. I brought the plant from Switzerland a few years ago and it lived well. I even managed to grow cuttings and passed them to a friend. However, my plant has died in the meantime whereas my friend still manages to grow cuttings successfully. She passed me back a quite big plant. Now I'm trying to grow cuttings. I've cut one branch that had brown roots (not sure if they are really roots as they don't grow in the soil but above) and put it in soil but it died a few days later. Then I cut another branch and put it in a cup of water in a dark spot in the house to grow roots. But roots never grew and the branch died too.
Singapore may not be the best place to grow rosemary but how come I've managed to do so before and some of my friends still do it now? Could you please tell me the best way I can grow it here in this climate. Also, how do I plant from cuttings, where do I cut and how long do I need to put it in water. Lastly, how do I preserve the plant I still have and prevent its death.
Your response is greatly appreciated as I'm trying to have a little European garden in Asia! Most herb literature are written for temperate climates and not for tropical!

Rosemary has a hard time surviving the heat during the hottest part of your year. Also it likes a cool rest period for the "winter". That is why it is rated as only surviving out of doors in zones 8 to 10 but not in the tropics.
However there is no reason why you should not be able to make it survive in a pot and bring it indoors when it gets too hot and for its rest period - I presume your house is air-conditioned.
You will have to be especially diligent in not letting the pot dry out completely at any time, because in your heat that happens very quickly. You might consider a white plastic pot to get maximum moisture retention and heat reflection, or a large clay pot to get cooling of the roots from the evaporation of water through the pores in the pot. This is one circumstance where over-potting might be a good idea.
The brown "roots" that you saw on the surface of the soil, probably were just some dead branches. The needles had probably fallen off and they then looked like roots.
To get rosemary cuttings to root, try it during the coolest part of the year. Cut off a 7 cm piece, strip off the needles on the lower 2 cm, dip this bare piece into fresh rooting powder, stick into sterile soil in a sterile pot, dampen with water, drain and cover with clean plastic. Put in a north window out of direct sunlight.