∀ ( x n ) n ∈ N ⊂ D : lim n → ∞ x n = c ⇒ lim n → ∞ f ( x n ) = f ( c ). Augustin-Louis Cauchy defined continuity of y = f ( x ) In mathematical notation,
In contrast, the function M( t) denoting the amount of money in a bank account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.Ī form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.Īs an example, the function H( t) denoting the height of a growing flower at time t would be considered continuous. The latter are the most general continuous functions, and their definition is the basis of topology.Ī stronger form of continuity is uniform continuity. The concept has been generalized to functions between metric spaces and between topological spaces. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.Ĭontinuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. A discontinuous function is a function that is not continuous. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument.
This means that there are no abrupt changes in value, known as discontinuities. In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function.
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