Definition: A function f: A --> R is continuous at a point c Element A if, for a

Question

Definition: A function f: A --> R is continuous at a point c Element A if, for all epsilon > 0, there exists delta > 0 such that whenever |x - c| < delta (and x Element A) it follows that |f(x) - f(c)| < epsilon. If f is continuous at every point in the domain of A, then we say that f is continuous on A. Explain how to take the logical negation of the precise definition of continuity. Explain what the above definition means in plain English. How would you explain the concept of continuity to someone who has not studied advanced math?

Explanation / Answer

negation Logic a. the operator that forms one sentence from another and corresponds to the English not b. a sentence so formed. It is usually written --p, ~p, p or ¬p, where p is the given sentence, and is false when the given sentence is true, and true when it is false negation [n?'ga·sh?n] (mathematics) The negation of a proposition P is a proposition which is true if and only if P is false; this is often written ~ P. Also known as denial. Negation a philosophical category expressing a certain type of relationship between two consecutive stages or states of a developing object or process. Negation is a necessary part of development and of the struggle between opposites. In “its comprehension and affirmative recognition of the existing state of things,” the dialectic “includes, at the same time also, the recognition of the negation of that state, of its inevitable breaking up” (K. Marx, in K. Marx and F. Engels, Soch., 2nd ed., vol. 23, p. 22). As it develops, an object inevitably reaches the stage of its own negation: that is, it becomes qualitatively something other than itself. The negation of the old and the rise of the new constitute a chain without beginning or end. In this process, the developing object simultaneously becomes something other than itself and, in a certain sense, remains what it was. For example, youth negates childhood and is itself negated by maturity, which is negated by old age. At the same time, these are merely different stages in the development of the same person. This continual self-negation characterizes ongoing self-development in nature, society, and cognition. “True, natural, historical, and dialectical negation (taken formally) is precisely what constitutes the driving principle of all development—the splitting into antitheses, their struggle and resolution” (F. Engels, ibid., vol. 20, p. 640). Dialectical negation is, above all, creative negation, in which the old is not simply discarded and destroyed but “removed” (aufgehoben) and preserved in a new capacity. V. I. Lenin emphasized that an essential part of the dialectic is “negation as a moment of connection, as a moment of development, retaining the positive” (Poln. sobr. soch, 5th ed., vol. 29, p. 207). This “retention” of the positive, the unity of negation and continuity in development, is an important feature of dialectical negation as a universal principle of being, which manifests itself in different ways at different levels of the organization of matter.

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