Delta of a down-and-out call against a vanilla

How does the delta of a down-and-out call (barrier below the current price, at or below the strike) compare with the delta of the plain European call it is built from?

The down-and-out delta is larger. The vanilla equals down-and-out plus down-and-in, and the down-and-in loses value as the stock rises (knock-in becomes less likely), so its delta is negative. Hence $\Delta_{\text{DO}} = \Delta_{\text{vanilla}} - \Delta_{\text{DI}} > \Delta_{\text{vanilla}}$, and near the barrier, where the option's value collapses toward zero, it can even exceed $1$.