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Continuous Function: Domains
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If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions
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Most of the elementary functions which have been introduced so far are continuous wherever they are defined. Hence the following types of functions are continuous at every number in their domains: root functions, polynomials, rational functions, trigonometric functions, and exponential functions. Moreover, the inverse function of any continuous function is continuous.
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All basic elementary functions are continuous at all points of their domains of definition. An important property of continuous functions is that their class is closed under the arithmetic operations and under composition of functions. More accurately, if two real-valued functions and , , are continuous at , then so is their sum , difference and product , and when ... their quotient (which is necessarily defined in the intersection of with a certain neighbourhood of ). If, as before, is continuous at and , , is such that , so that the composite makes sense, if there is a such that and if is continuous at , then is also continuous at . Thus, in this case
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