Two models of opinion dynamics are entangled in order to build a more realistic model of inflexibility. The first one is the Galam unifying frame (GUF), which incorporates rational and inflexible agents, and the other one is the Continuous Opinions and Discrete Actions model. While initially in GUF, inflexibility is a fixed given feature of an agent, it is now the result of an accumulation for a given agent who makes the same choice through repeated updates. Inflexibility thus emerges as an internal property of agents becoming a continuous function of the strength of its opinion. Therefore, an agent can be more or less inflexible and can shift from inflexibility along one choice to inflexibility along the opposite choice. These individual dynamics of the building up and falling off of agent inflexibility are driven by the successive local updates of the associated individual opinions. New results are obtained and discussed in terms of predicting outcomes of public debates.