I came across some graphs depicting the delta of a down-and-out call. They show that, if the risk free rate of return is 0, the delta is constant at 1. However, if the rate of return is for example 5%, the delta rises as the stock price approaches the barrier. I can't figure out why.
You can find an accurate delta graph on page 62 of the following document:
What you wrote is definitely incorrect. With a down and out call delta drops as the stock price approaches the barrier, it reaches zero smoothly as it approaches the barrier for close to expiration options and exhibits much more of discontinuous jump from values around 0.5 down to 0 at the barrier level for longer dated options.
I am not sure if any numeric comment about the dynamics could be made without knowing the relative strike and the barrier. In general, however, you would expect delta to approach zero as you are approaching the barrier and delta approaching one as you go away from the barrier and deep into the money. In general, CDO and PUO are fairly easy to manage, since as the underlying approaches the barrier, the delta is already fairly low and the discontinuity is small. For obvious reasons, the discount to vanilla is fairly small and these products trade far less frequently then CUO/PDO and PDI/CUI.