Modern information and communication technologies used by smart grids are subject to cybersecurity threats. This paper studies the impact of integrity attacks on real-time pricing (RTP), a key feature of smart grids that uses such technologies to improve system efficiency. Recent studies have shown that RTP creates a closed loop formed by the mutually dependent real-time price signals and price-taking demand. Such a closed loop can be exploited by an adversary whose objective is to destabilize the pricing system. Specifically, small malicious modifications to the price signals can be iteratively amplified by the closed loop, causing inefficiency and even severe failures such as blackouts. This paper adopts a control-theoretic approach to deriving the fundamental conditions of RTP stability under two broad classes of integrity attacks, namely, the scaling and delay attacks. We show that the RTP system is at risk of being destabilized only if the adversary can compromise the price signals advertised to smart meters by reducing their values in the scaling attack, or by providing old prices to over half of all consumers in the delay attack. The results provide useful guidelines for system operators to analyze the impact of various attack parameters on system stability, so that they may take adequate measures to secure RTP systems.