On Continuity (2010-2019)

"Today, most mathematicians have embraced the axiom of choice because of the order and simplicity it brings to mathematics in general. For example, the theorems that every vector space has a basis and every field has an algebraic closure hold only by virtue of the axiom of choice. Likewise, for the theorem that every sequentially continuous function is continuous. However, there are special places where the axiom of choice actually brings disorder. One is the theory of measure." (John Stillwell, "Roads to Infinity: The mathematics of truth and proof", 2010) 

"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)

"Continuity is the rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable - say x - is associated with a value of a dependent variable - say y. Continuity of a function is sometimes expressed by saying that if the x-values are close together, then the y-values of the function will also be close. But if the question 'How close?' is asked, difficulties arise." (Erik Gregersen [Ed.], "Math Eplained: The Britannica Guide to Analysis and Calculus", 2011)

"If we wish the word ‘continuous’ to prohibit jumps in a function, its definition must somehow control the vertical change of the function at a sort of microscopic level. That is, at any point on a ‘continuous’ function, the nearby points ought to be as ‘close’ as possible." (David Perkins, "Calculus and Its Origins", 2012)

"A coordinate system then is a map from a region of the differentiable manifold to the set of n-tuples. If we choose a point, the map determines an n-tuple of coordinates of the point. Hence, we have the coordinate map and the coordinates as components, or values taken by the set of coordinate functions that constitute the map. When two coordinate systems overlap, it is required that the functionsexpressing the coordinate transformation are continuous and have continuousderivatives up to some order, appropriate to achieve some specific purpose." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Charts play the role of providing the arena that supports differentiation in a continuous set of points. The purpose of having different charts and not just one is to be able to adequately cover all points in the set, one small piece at the time. We then endow the set with additional structure, materialized through a system of connection equations and a system of equations of structure, the latter speaking of whether the former is integrable or not." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"There is a great virtue in calculus of differential forms. It is appropriate for differentiation in general spaces known as differentiable manifolds, which, in general, lack a metric and a connection or rule to compare vectors at different points. We must, however, have a concept of continuity in the space in question, which eliminates automatically discrete sets of points. The set must be such that regions of the same can be represented unequivocally by open sets of n-tuples of real numbers (it could also be complex numbers [...]). The openness of the set has to do with the behavior at the borders of the region." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Sometimes mathematical advances happen by just looking at something in a slightly different way, which doesn’t mean building something new or going somewhere different, it just means changing your perspective and opening up huge new possibilities as a result. This particular insight leads to calculus and hence the understanding of anything curved, anything in motion, anything fluid or continuously changing." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Continuity is only a mathematical technique for approximating very finely grained things. The world is subtly discrete, not continuous." (Carlo Rovelli, "The Order of Time", 2018)

"It is not possible to think of duration as continuous. We must think of it as discontinuous: not as something that flows uniformly but as something that in a certain sense jumps, kangaroo-like, from one value to another. In other words, a minimum interval of time exists. Below this, the notion of time does not exist - even in its most basic meaning." (Carlo Rovelli, "The Order of Time", 2018)