In reality, prices cannot be adjusted continuously.
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On the demand side, past prices may affect the firms' current goodwill through consumers' learning about the good or switching costs. On the supply side, past prices affect current inventories.
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The presence of price rigidities raises the possibility that price reactions are not bootstrap reactions but are simply attempts to regain or consolidate market share.

(Tirole (1988), p.253-4)

In Maskin and Tirole (1988), they assume two firms choose their prices asynchoronously, at odd (even) periods, firm 1 (2) chooses its price. They consider Markov strategies, the simple pricing strategies that depend only on the "payoff-relevant information", and look for a perfect equilibrium in which the firms use Markov strategies. (They call it "Markov perfect equilibrium")

Despite the restriction to simple (Markov) strategies, multiple equilibria exist (indeed, there also exist several kinked-demand-curve equilibria). However, it can be shown that in any Markov perfect equilibrium, profits are always bounded away from the competitive profit (which is 0).
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The intuition here is that if firms were stuck in the competitive price region, with the prospects of small profits in the future, a firm could raise its price dramatically and lure its rival to charge a high price for at least some time (the rival would not hurry back to nearly competitive prices). Thus tacit collusion is not only possible (as in the supergame approach) but necessary. Furthermore, it can be shown that there exists only one pair of equilibrium strategies that sustain industry profits close to the monopoly profit. These strategies from a symmetric kinked-demand-curve equilibrium at the monopoly price, and they are the only symmetric "renegotiation-proof" equilibrium strategies (whatever the current price, the firms cannot find an alternative Markov perfect equilibrium that they both prefer).

(Tirole (1988), p.256)