Wondering how you would think about the following thought experiment - suppose you sell an OTM call option and plan to implement a delta hedging strategy whereby if the price of the stock were to increase and reach the strike, you hedge 100% by buying the shares at the strike price, and then subsequently if it were to come back down you take off the hedge - i.e. the hedge is always 100% or 0%. Assuming you could do this without any transaction costs, then you could just collect the premium from selling the option and guarantee you will be hedged at expiry.
There must be a flaw in this logic as it would imply a free collection of premium, so I am wondering where that flaw is and how it might relate to discrete vs. continuous time hedging.