Non-standard analysis is a branch of mathematics that extends the traditional framework of calculus by introducing the concept of infinitesimals. Infinitesimals are quantities that are closer to zero than any standard real number but are not zero themselves. This approach allows mathematicians to rigorously work with concepts that were previously intuitive, such as limits and continuity, in a more straightforward manner. Developed by Abraham Robinson in the 1960s, non-standard analysis provides tools for analyzing functions and sequences using these infinitesimal values. It has applications in various fields, including mathematical logic, model theory, and even physics, where it can simplify complex problems by providing a different perspective on continuity and change.