Flaw in hedging a call by buying above the strike

A trader has sold an out-of-the-money call. He says he can hedge it at no cost: whenever the stock rises through the strike he buys one share, and whenever it falls back through the strike he sells it, so at expiry he holds a share exactly when the call is in the money. What is wrong with this argument?

He cannot trade exactly at the strike, and the small losses on each crossing do not vanish. In practice he buys at $K+\varepsilon$ and sells at $K-\varepsilon$, losing about $2\varepsilon$ per round trip. A diffusion that hits $K$ recrosses it infinitely often in any short interval, so as $\varepsilon\to0$ the number of crossings grows like $1/\varepsilon$ and the expected total cost stays positive; it equals the option's value (the local time at the strike).