The greatest contribution of Wilstras in the field of mathematical analysis is that in the rigorous trend of mathematical analysis initiated by Cauchy and Abel, the foundation of real analysis and complex analysis is systematically established in ε -δ language, and the analysis is basically arithmetically completed. He introduced the concept of uniform convergence, thus expounding the theorem of item-by-item differential and item-by-item integral of function series.
In the process of establishing the analysis foundation, a series of topological concepts in real number axis and n-dimensional Euclidean space are introduced, and Riemann integral is extended to discontinuous functions on countable sets. In 1872, Weierstrass gives the first example of a function that is continuous everywhere but differentiable everywhere, which makes people realize the difference between continuous and differentiable, and thus leads to the study of a series of abnormal functions such as peano curve.
Hilbert commented on him like this: "Wilstrass laid a solid foundation for mathematical analysis with his critical spirit and profound insight. By clarifying the concepts of minimax, minimax, function and derivative. He eliminated all kinds of wrong formulas that still appeared in calculus, sorted out all kinds of confusing concepts about infinity and infinitesimal, and finally overcame the difficulties arising from the fuzzy thoughts of infinity and infinitesimal. Today, analytical science can reach such a harmonious, reliable and perfect level, which is essentially due to the scientific activities of Wilstrass. "