Analytic continuation is a mathematical technique used to extend the domain of a given analytic function beyond its original limits. An analytic function is one that can be represented by a power series in a neighborhood of every point in its domain. By finding a new function that agrees with the original function in a certain region, mathematicians can explore values where the original function may not be defined. This method is particularly useful in complex analysis, where functions of complex variables are studied. It allows for the exploration of properties of functions like the Riemann zeta function or complex logarithm, which may have singularities or undefined points in their original forms.