What is this fruit called?
It looks like either Pepino Melon (which I'm growing this year, and I tried to grow in 2015) or Tzimbalo, by the color, the dark striping and the leaves. Eggplant is related to both, but eggplant leaves leaves are broader and bigger than I see in the picture, usually (if not always). Pepino Melons are typically (not always) bigger and longer than Tzimbalo, but it's hard to tell from your picture which it is. I personally think it looks more like a Pepino Melon by the color, lack of sheen, and the way the leaves look.

It looks like the plant might have spider mites, by the speckling on the foliage. Mine had spider mites in 2015. If you do have spider mites, showering the plant with a shower nozzle on your hose every two or three days may help, at least if you're in a semi-arid to arid climate.

Pepino Melons are not melons, although some people think they taste like them. They're in the Solanum genus of the nightshade family.