Continuous Function

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A Continuous Function is a real-valued function for which small delta in the function domain results in a small delta in the function range.



  • (Wikipedia, 2014) ⇒
    • In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

      Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.

      As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

    • A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".

      There are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the definitions below, :[math]\displaystyle{ f\colon I \rightarrow \mathbf R. }[/math] is a function defined on a subset I of the set R of real numbers. This subset I is referred to as the domain of f. Some possible choices include I=R, the whole set of real numbers, an open interval :[math]\displaystyle{ I = (a, b) = \{x \in \mathbf R \,|\, a \lt x \lt b \}, }[/math] or a closed interval :[math]\displaystyle{ I = [a, b] = \{x \in \mathbf R \,|\, a \leq x \leq b \}. }[/math] Here, a and b are real numbers.