The chance of choosing 0.5 from the interval 0 to 1 is 0. So if you want the change of choosing between 0.4 and 0.6 in that interval, you can't just sum all chances between 0.4 and 0.6 you'd have infinity x 0 which isn't 0.2 .

Instead, to deal with this 'a lot of really small things' we need a slightly different approach. Then draw a pdf, and talk about area under the curve. Perhaps relate this to the discrete case where the curve is just really blocky.