I have trouble understanding solution of the Drunkard's walk problem. Here is the original statement of the problem:
There once was a drunk man who wandered far too close to a cliff. From where he stands, one step forward would send the drunk man over the edge. He takes random steps, either towards or away from the cliff. At any step, his probability of taking a step away is 2/3 and a step towards the cliff is 1/3. What is his chance of escaping the cliff?
From the book that I read, a solution involves proving continuity of some function. Here's a relevant portion of the solution:
What I don't understand is, why continuity comes into play here? Solution from other sources (example) that also involve solving for P1 don't mention continuity at all. It seems to me that the author really stresses the importance of continuity, whose proof he skips by the way ("beyond the scope of this book").