In this study, we consider a family of uni-parametric linear optimization problems that the objective function, the right, and the left hand side of constraints are linearly perturbed with an identical parameter. We are interested in studying the effect of this variation on a given optimal solution and the behavior of the optimal value function on its domain. This problem has several applications, such as in linear time dynamical systems. A prototype example is provided in dynamical systems as a justification for the practicality of the study results. Based on the concept of induced optimal partition, we identify the intervals for the parameter value where optimal induced partitions are invariant. We show that the optimal value function is piecewise fractional continuous in the interior of its domain, while it is not necessarily to be continuous at the endpoints. Some concrete examples depict the results of the analysis.