LYCOS RETRIEVER 
Continuous Function: Interval
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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 f(x) = 1/x and g(x) = (sin x)/x. Neither function is defined at x = 0, so each has domain R\{0} of real numbers except 0, and each function is continuous. The question of continuity at x = 0 does not arise, since it is not in the domain. The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of g at 0 is 1, g can be extended continuously to R by defining its value at 0 to be 1.
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Many properties of real-valued continuous functions of a single variable carry over to continuous mappings between topological spaces. A generalization of Weierstrass' theorem mentioned above: The continuous image of a compact topological space in a Hausdorff space is compact. A generalization of Cauchy's intermediate value theorem for a continuous function on a closed interval: A continuous image of a connected topological space in a topological space is ... connected. A generalization of the theorem on the inverse function of a strictly monotone continuous function: A continuous one-to-one mapping of a compactum onto a Hausdorff space is a homeomorphism. A generalization of the theorem on the limit of a uniformly-convergent sequence of continuous functions: If is a uniformly-convergent sequence of mappings of a topological space into a metric space that are continuous (at a point ) then the limit mapping is also continuous (at ). A generalization of Weierstrass' theorem on the approximation of functions that are continuous on a closed interval is the Stone–Weierstrass theorem.
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The inner product of a continuous function with itself has an important interpretation in many physical situations. For example, Ohm's law of electrical circuits states that the power dissipated by a resister is given by the squared voltage divided by resistance. If v(t) describes the time course of voltage across a 1 ohm resistor, then the time-course of power consumption is v2(t) and the total amount of energy consumed over the interval from 0 to T seconds is
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A function that is continuous on an interval has no gaps and hence cannot "skip over" values. If a function is continuous on a closed interval from x = a to x = b, then it has an output value for each x between a and b. In fact, it takes on all the output values between f (a) and f (b); it cannot skip any of them. More formally, the Intermediate Value Theorem says:
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A continuous function on a closed interval has a maximum and a minimum, and assumes all values between them. For example, if f(a)×f(b) < 0, then there must be at least one point c in [a, b] where f(c) = 0.
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Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the series a n cos (b n x7r), where a is positive and less n=0 than unity, and b is an odd integer exceeding (1+27r)/a. It can be shown that this series is uniformly convergent in every interval, (ii.) and that the continuous function f(x) represented by it has the property that there is, in the neighbourhood of any point xo, an infinite aggregate of points x', having xo as a limiting point, for which { f (x') - f (xo) }/(x' - x 0 ) tends to become infinite with one sign when x' - xo approaches zero through positive values, and infinite with the opposite sign when x' - xo approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a function f(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F'(x)=f(x); and, if f(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval... small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction.
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