Limits and continuity (mathematics)
Limits and continuity are foundational concepts in mathematics, particularly in calculus, analysis, and topology. The notion of a limit refers to the value a function approaches as its input approaches a specific point. This concept has roots in ancient civilizations, where mathematicians like Archimedes and Liu Hui explored practical and theoretical uses of limits, such as calculating the circumference of circles. Continuity, on the other hand, describes a function's behavior in preserving closeness, meaning small changes in input result in small changes in output.
These ideas gained clarity and rigor in the 17th century with the independent development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. Later, Augustin-Louis Cauchy formalized the definitions of limits and continuity, establishing them as essential to modern calculus. The contemporary definition of a limit involves precise ε-δ conditions to verify its existence, while continuity at a point is defined through limits of function values relative to the function's output. These concepts not only play a crucial role in theoretical mathematics but also have practical applications in fields such as finance and physics, where they help model continuous phenomena. Today, the exploration of limits extends to sequences and geometric spaces, with ongoing developments in non-standard analysis introducing novel perspectives on infinitesimals and continuity.
Limits and continuity (mathematics)
Summary: One of the key concepts of calculus, the limit is the value a function approaches as its input approaches a given value.
The concepts of limit and continuity are fundamental in calculus, analysis, and topology. Their inception can be traced back more than 2000 years to Greece, China, Babylon, Egypt, and other places. During the inception of calculus, introduced independently by Isaac Newton and Gottfried Wilhelm Leibniz, these concepts were still vague and controversial. In the twenty-first century, these concepts are explored in high school. The modern limit of a function f(x) is the value the function tends to when x changes in a structured way; for instance, x approaches a specific value. Many different definitions of limits in calculus can be combined into a single definition of limit in topology. Moreover, a function is continuous if it preserves closeness or, equivalently, if it preserves limits, and topology is the study of continuous functions and the properties they preserve. These ideas underpin many mathematical results and can be used to organize and simplify mathematical processes. Limits are also useful in real life to understand such concepts as demographics, finance, or terminal velocity. Modeling discrete data with a continuous function and the notion of continuous payments are also important. Leibniz defined a principle of continuity, or lex continuitatis, which inspired philosophers such as Charles Peirce. The notions of limit and continuity are still debated philosophically, as in whether growth spurts are continuous over time.
The Ideas of Limit and Continuity in the Ancient World
The idea of limit in the ancient world was related mainly to two activities: one was more practical, like measuring length, area, and volume, and the other was more abstract, such as making sense of numbers that are not rational. For example Archimedes from Greece and Liu Hui in China used regular polygons, inscribed in a circle, increasing the number of sides of the polygons, in order to compute the length of the circumference and the area of a circle. In the process, approximations of π were computed. Eudoxus from Greece created his theory of proportions to legitimize irrationals like √2.
This theory is expounded in Book Five of Euclid’s Elements. It is also a precursor of the contemporary theory of the real numbers. Ancient mathematicians, such as Zeno of Elea and Aristotle, wrestled with the notion of continuity. They debated whether motion, time, and space are continuous. The paradox about Achilles and the tortoise illustrated the interplay between the ideas. The paradox states that Achilles can never overtake the tortoise if the tortoise is given a head start, because by the time Achilles reaches its initial position the tortoise has farther advanced and so on; infinitely many segments of time are necessary.
The Calculus of the Infinitesimals
In the middle of the seventeenth century after significant advances in science, particularly in physics, mechanics, and geometry, the methods of infinitesimal calculus were introduced independently by Isaac Newton in England and by Gottfried Wilhelm Leibniz in Germany. Newton assumed that geometric magnitudes are generated by continuous motion, and some historians suggest that he may have been the first to present a limit argument using an infinitesimal like epsilon. Leibniz explored a principle of continuity but is not thought to have explored the derivative as a limit. He viewed the ocean as continuous.
The quest to find an acceptable, rigorous foundation for the new calculus was ongoing. There were attempts made to follow the method of exhaustion from the Greeks or to use series instead of infinitesimals to introduce the derivative. Jean le Rond d’Alembert stressed the importance of a firm foundation for limits and explored a geometric limit of secant lines. But it was Augustin-Louis Cauchy who introduced contemporary definitions of limit and continuity and placed them as a cornerstone of calculus. It is interesting to mention that another mathematician, Bernard Bolzano, from Prague and a contemporary of Cauchy, came up with the same definitions first, but he was more isolated and his work did not get the same recognition that the Cours d’analyse enjoyed. The German mathematician Karl Weierstrass solidified the rigorous definitions and made significant contributions to the development of analysis.
Contemporary Definitions
The limit of a function is a dynamic concept. The input of the function varies and the output varies as well. Intuitively speaking, one says that limx→cf(x)=L when the values of f become closer and closer to the number L when c gets closer and closer to c. This intuitive concept is easy to grasp and also not difficult to observe if one has a graph of the function f. But this should be expressed rigorously, so that there is a tool to verify whether the limit exists.
Definition: The limx→cf(x)=L if and only if for each ε>0 there is a δ>0, such that if 0<|x-c|<δ, then |f(x)-L<ε.
This definition enables mathematicians to verify the existence of limit or to make an argument that there is no limit and is also a tool to prove many properties about limits. The number c does not have to be in the domain of the function, but one should be able get δ close to it from the domain for any positive δ. The concept of limit is used to define a continuous function.
Definition: A function f is continuous at a point c if c is in the domain of the function and for any positive ɛ there is a positive δ, such that if |x-c|<δ, then |f(x)-f(c)|<ε.
The concept of limit is used to define the definite integral and to measure area.
Contemporary Developments.
Limits of other objects, such as sequences and geometric spaces, can be defined and are important in many disciplines of mathematics and continuity is still explored in the field of topology. One twentieth-century development that goes back to the history of limits occurred around 1960 when Abraham Robinson entertained the idea that the advantages of infinitesimal calculus can be utilized as soon as the infinitesimals are defined in a rigorous way. This would eliminate the use of limit in the way it is known and make the analysis very much like algebra, as soon as the number system is extended to permit infinitely small and infinitely large numbers. This is exactly what Robinson did using tools from logic, a development called “non-standard analysis.”
Bibliography
Edwards, Charles. The Historical Development of Calculus. New York: Springer, 1994.
Ilarregui, Begoña, and J. Nubiola. “The Continuity of Continuity: A Theme in Leibniz, Peirce, and Quine.” In Leibniz und Europa, VI. Hannover, Germany: Gottfried-Wilhelm-Leibniz-Gesellschaft, 1994.