On the Continuity of Functions and Their Application in Polynomial Approximation

Researchers continue to explore the topological properties of continuity, including the use of open sets, neighborhoods, and limit points to define and characterize continuous functions. For other aspects, scholars have made achievements in both linear operators and functionals in the context of functional analysis, including applications to integral and differential equations. The development would also see achievements in studies with partial derivatives, directional derivatives, and differentiability. This study primarily discusses how to understand the continuity of one-variable and multivariable functions and introduces some methods for determining continuity. It then explores the application of function continuity in polynomial approximation, specifically focusing on approximating continuous functions in different scenarios using polynomials.