n of the true extent of the general concepts of analysis is typical of the theory of functions of a real variable. (At the very start of the theory’s development B. Bolzano and later Weierstrass found, for example, that a continuous function may not have a derivative at any point.) Investigations in the theory of functions of a real variable led to general definitions of the concepts of the measure of a set, measurable functions, and integral, which play an important role in modern mathematics. The foundations of the modern theory of functions of a real variable were laid by mathematicians of the French school (Jordan, Borel, H. Lebesgue, R. Baire), and later the leading role was assumed by the Russian and Soviet school.
In addition to its immediate interest, the theory of functions of a real variable greatly influenced the development of many other branches of mathematics. Its methods proved to be essential in the construction of the foundations of functional analysis. Although the methods of functional analysis developed under the influence of the theory of functions of a real variable and set theory, the content and nature of the problems of functional analysis border directly on classical analysis and mathematical physics. Functional analysis is virtually indispensable (chiefly in the form of the theory of operators) in quantum physics. The first conscious isolation of functional analysis as a special branch of mathematics was made by V. Volterra in the late 19th century. The calculus of variations and the theory of integral equations, whose systematic development was also begun by Volterra and continued by E. Fredholm, were now perceived as parts of functional analysis. The most important special case of operators in Hilbert space, whose primary role was brought out in the works of Hilbert on integral equations, was analyzed with particular intensity.
Most of the mathematical problems posed by natural science and technology reduce to the solution of differential equations, both ordinary (in the study of systems having a finite number of degrees of freedom) and partial differential equations (in the study of continuous media and in quantum physics). Therefore, all directions of investigations of differential equations were intensively developed during this period. The methods of operational calculus were created to solve complex linear systems. The method of parametric expansion was used widely in the study of nonlinear systems having small nonlinearity. The analytic theory of ordinary differential equations continued to be developed (Poincare and others). However, in the theory of ordinary differential equations, problems of the qualitative investigation of their solutions now attracted the greatest attention: the classification of singular points (Poincare and others) and problems of stability, which were studied in great depth by A. M. Liapunov.
The qualitative theory of differential equations served for Poincaré as a starting point for a broadly conceived continuation of the investigations, barely outlined by Riemann, into the topology of manifolds, particularly the study of the fixed points of continuous mappings of manifolds onto themselves. The “combinatorial,” “homologous,” and “homotopic” methods of modern topology originated here. Another direction arose in topology based on set theory and functional analysis and led to the systematic development of the theory of general topological spaces.
The theory of partial differential equations assumed an essentially new form at the end of the 19th century owing to intense interest in boundary value problems and the elimination of the restriction to analytic boundary conditions. The analytic theory, which originated with Cauchy, Weierstrass, and S. V. Kovalevskaia, did not lose its importance in the process. However, it recedes somewhat into the background, since it was found that it did not guarantee correctness in the solution of boundary value problems, that is, it did not guarantee the possibility of finding an approximate solution if the boundary conditions were