Continuidad en varias variables: Análisis riguroso y el duelo entre 4 inteligencias artificiales

In this video, I tackle a classic problem on the continuity of functions of several variables, rigorously analyzing the function's behavior at a critical point and studying the different approximation paths, as required in a formal treatment of multivariable calculus. The goal is not only to determine whether the function is continuous or not, but also to deeply understand the geometric interpretation of the limit in several variables and the relationship between algebraic analysis and its graphical representation in space. Furthermore, I decided to test four artificial intelligences: DeepSeek, ChatGPT, Claude, and Gemini. I presented them with the same problem to evaluate: The mathematical soundness of their arguments. The clarity of their limit analysis. The rigor of their justification. And, especially, their ability to provide a suitable graphical representation that accurately reflects the geometric interpretation of the problem. The results were surprising. Each offered distinct approaches, with clear strengths and weaknesses. In the video, I analyze their solutions, compare their reasoning, and show which one achieves the best conceptual construction and which one presents the most accurate graphical representation of the phenomenon. If you're interested in rigorous multivariable calculus, the real geometric interpretation of limits in ℝ², and want to see how far AI has come in solving formal mathematical problems, you'll find this analysis very interesting. Subscribe and join me in this serious and uncompromising comparative study.