 Solution

A down-and-out digital call with strike at the barrier pays one exactly when the barrier is avoided until expiry. Reflection therefore gives its alive priceThe complementary contract pays one at expiry when the barrier has been hit. Its price isThis is a cash-at-expiry one-touch option, not cash paid at the first touch. After the barrier has already been hit, its price is $e^{-r\tau }$ regardless of the current spot. In particular this branch applies immediately for $S\le B$. Current spot alone does not encode the full path history.