In the preceding chapter, we introduced the concept of a well-posed problem in the sense of Hadamard. We also observed that certain Cauchy problems do not fall under the category of well-posed problems. We realized that these problems might lack continuous dependence of solutions on the initial data. This phenomenon poses a significant challenge in the analysis of problems arising from real-world applications. In practical situations, data measurements are inherently subject to approximation errors. Consequently, the impact of these errors must always be considered to ensure that theoretical results remain applicable. Importantly, non-continuous dependence can also arise in simpler cases, such as approximating derivatives or estimating velocity from a time function. A small error in the time function can cause a larger error in the velocity, illustrating that a well-approximated time function may not yield a good approximation of the velocity. This example highlights the issue that the operator from time functions to their derivatives may not be continuous, as small errors in the input can result in large discrepancies in the output. Throughout the discussion, when referring to a problem as ill-posed in the sense of Hadamard, we imply that while solutions may exist and be unique, the problem lacks continuous dependence on the initial data, and this should be understood in the context of the specific applied problem.