In the real-world optimization problems, continuous uncertainties (such as field uncertainty and demand uncertainty) are usually unbounded and have unknown probability density functions, which have significant influences on the fluctuation in production. As the classical robust counterpart optimization approach is mainly for dealing with bounded uncertainties, the concept of "flexible uncertainty set" is proposed on the basis of the definitions of the classical box, ellipsoidal, and polyhedral uncertainty sets. Next, the continuous uncertainties can be transformed into bounded ones, which are connected with a predefined "confidence level". Moreover, their corresponding robust formulations and a priori probability bounds of constraints' violation are derived and proven in detail. Several numerical examples (including a real-world process industry example) are introduced to illustrate the new formulations induced by flexible uncertainty sets, which have been proven less conservative and with tight probability bounds of constraints' violation.