"At the large scale where many processes and structures appear continuous and stable much of the time, important changes may occur discontinuously. When only a few variables are involved, as well as an optimizing process, the event may be analyzed using catastrophe theory. As the number of variables in- creases the bifurcations can become more complex to the point where chaos theory becomes the relevant approach. That chaos theory as well as the fundamentally discontinuous quantum processes may be viewed through fractal eyeglasses can also be admitted. We can even argue that a cascade of bifurcations to chaos contains two essentially structural catastrophe points, namely the initial bifurcation point at which the cascade commences and the accumulation point at which the transition to chaos is finally achieved." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)
"It would seem that there may be a possible synthesis that can de-bifurcate bifurcation theory and give a semblance of order to the House of Discontinuity. Reasonable continuity and stability can exist for many processes and structures at certain scales of perception and analysis while at other scales quantum chaotic discontinuity reigns. It may be God or it may be the Law of Large Numbers which accounts for this seemingly paradoxical coexistence." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)
"Mathematically there are clearly degrees of continuity. Thus a function may be kinked, itself continuous but possessing discontinuous first derivatives. Or a discontinuity may occur in a higher, more remote derivative, but not in lower order ones." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)
"The major strength of catastrophe theory is to provide a qualitative topology of the general structure of discontinuities. Its major weakness is that it frequently is not associated with specific models allowing precise quantitative prediction, although such are possible in principle." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)
"We must recognize here that continuity and discontinuity may not always be clearly distinguishable. Ultimately what may matter is the precision of the perspective applied to the question at hand. Thus, if quantum mechanics is true, then reality is fundamentally discrete at a microscopic level. Nevertheless it may be useful to view it as continuous when the analysis is less precise, much as we perceive a movie to be continuous even though it actually consists of a sequence of discrete and discontinuous still photographs. Our mind draws an in-visible line between the stills creating the illusion of continuity." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)
"Symmetry is bound up in many of the deepest patterns of Nature, and nowadays it is fundamental to our scientific understanding of the universe. Conservation principles, such as those for energy or momentum, express a symmetry that (we believe) is possessed by the entire space-time continuum: the laws of physics are the same everywhere." (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)
"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"A homeomorphism may be thought of as the best possible type of continuous function, and homeomorphic spaces are considered the same in topology. [...] A complex has two structures: that of the topological space underlying the complex, and the subdivision of the complex into cells. The corresponding functions will have to preserve this dual nature. In particular, it would be nice if these functions, yet to be defined, would also induce nice functions on the homology of the complex. Since homology has an algebraic group structure, we want the functions to induce homomorphisms on the homology groups. The best of all functions would be homeomorphisms on the underlying topological space and also isomorphisms oii the homology groups." (L Christine Kinsey. "Topology of Surfaces", 1993)
"In any mathematical study, the first thing to do is specify the type or category of objects to be investigated. In topology, the most general possible objects of study are sets of points with just enough structure to be able to define continuous functions." (L Christine Kinsey. "Topology of Surfaces", 1993)
"In mathematics, a function is a sort of converter: It converts a value to some other value. […] the mathematical definition of a continuous function requires that any small change in the input will result in a small change to the output." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)
"Two objects are topologically identical if there is a continuous deformation (to be more precisely defined later) from one to the other. The bending and stretching [...] are examples of such continuous deformations. Thus, topology is sometimes called the study of continuity, or, a more hackneyed term, 'rubber-sheet geometry', as one pretends everything is formed of extremely flexible rubber." (L Christine Kinsey. "Topology of Surfaces", 1993)
"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)
"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"The property of smoothness includes the property of continuity. The notion of a topological space was born from the development of abstract algebra as a universal notion for the property of continuity." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"When we study the concept of continuity by itself, numbers are not necessary as long as you are dealing with objects which have the property of 'nearness'. Therefore, if we can introduce the notion of 'nearness' detached from numbers from a purely abstract point of view, then we can discuss topics related to continuity based upon this notion. This approach enables us to become familiar with the science we call mathematics." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"Topology studies those characteristics of figures which are preserved under a certain class of continuous transformations. Imagine two figures, a square and a circular disk, made of rubber. Deformations can convert the square into the disk, but without tearing the figure it is impossible to convert the disk by any deformation into an annulus. In topology, this intuitively obvious distinction is formalized." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
"Two figures which can be transformed into one other by continuous deformations without cutting and pasting are called homeomorphic. […] The definition of a homeomorphism includes two conditions: continuous and one- to-one correspondence between the points of two figures. The relation between the two properties has fundamental significance for defining such a paramount concept as the dimension of space." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)
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