  Rounding border effects at the escape point of open integrable billiards are
analyzed via the escape times statistics and emission angles. The model is the
rectangular billiard and the shape of the escape point is assumed to have a
semicircular form. Stickiness and self-similar structures for the escape times
and emission angles are generated inside "backgammon" like stripes of initial
conditions. These stripes are born at the boundary between two different
emission angles but same escape times. As the rounding effects increase,
backgammon stripes start to overlap and the escape times statistics obeys the
power law decay and anomalous diffusion is expected. Tiny rounded borders
(around $0.1\%$ from the whole billiard size) are shown to be sufficient to
generate the sticky motion, while borders larger than $10\%$ are enough to
produce escape times with chaotic decay.
