Down-and-out versus down-and-in call

Plot against the current spot price (a) a call that pays out only if the stock never touches a barrier set below the current price, and (b) a call that pays out only if that barrier is touched. How are the two related?

They add up to the vanilla call. The knock-out is zero at and below the barrier and rises towards the vanilla value far above it; the knock-in equals the vanilla call at the barrier and decays towards zero as spot moves far above it. Exactly one of them pays on every path, so down-and-out plus down-and-in equals the ordinary call.