Continuity Over an Interval

6 Key excerpts on "Continuity Over an Interval"

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  eBook - ePub

  Mathematics for Enzyme Reaction Kinetics and Reactor Performance

  • F. Xavier Malcata(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  d.

  Figure 9.3

  Graphical representation of function f {x}, (a) continuous within interval]a,b[ or (b, c) discontinuous only at x = c ∈ ]a, b[ – with (b) and in the left and right vicinities, respectively, of c, and (a, c) in the vicinity of c – with (b) concomitant inexistence of and (c) .

  Mathematically speaking, a function f {x} is said to be continuous at point x = a if whichever sequence (a

  n

  ) converging to a implies that sequence ( f {a

  n

  }) converges to f {a}, i.e.

  (9.138)

  this statement implies that both and f {a} must exist, and further that they must be equal. A similar definition can be conceptualized for a multivariate function, although being of a lesser usefulness in practice. In view of Eq. (9.138) , one may resort to Eqs. (9.73) and (9.86)

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  CLEP® Calculus Book + Online

  • Gregory Hill, Mel Friedman(Authors)
  • 2013(Publication Date)
  • Research & Education Association

    (Publisher)

  x) all have vertical asymptotes. In addition, the greatest integer function has jump discontinuities at every integer in its domain.

  CONTINUITY OF A FUNCTION

  In the absence of a stated domain, a function is continuous if it is continuous at every point of its domain. Interestingly, this means that, for instance, is considered a continuous function since x = 0 is not in its domain, even though most people would see it as being discontinuous at its vertical asymptote. Fortunately, this approach to continuity of functions is rarely used in introductory calculus courses. Most often, one is asked to determine continuity over a stated domain, or determine points of discontinuity over the real numbers. In this context, is not continuous over the real numbers.

  EXAMPLE 2.12

  On the domain [–4, 4], where is discontinuous?

  SOLUTION

  Zeros in the denominator cause discontinuities, so f(x) is discontinuous at x = 2 and x= –2.

  It is also important to note that any algebraic combination of continuous functions is also a continuous function with the exception of division by a function with a value of 0.

  If g and h are continuous at a, then the following algebraic combinations of functions are continuous at a:

  1) g±h

  2) g.h

  3) if h(a) ≠ 0

  Additionally, a composition of continuous functions is also continuous.

  If g is continuous at a, and h is continuous at g(a), then h(g(x)) is also continuous at a.

  One of the most important immediate consequences of continuity is the Intermediate Value Theorem. We state it without proof.

  Intermediate Value Theorem

  Let f be continuous on the closed interval [a, b], and let M be any number between f(a) and f(b). There exists a number c in (a, b) such that f(c) = M.

  Informally, this theorem says that on a given interval [a, b], a continuous function will, at least once, take on all y-values between f(a) and f(b

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  Encyclopedia of Financial Models

  • Frank J. Fabozzi, Frank J. Fabozzi(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  x ∈ [1, 2).

  Note that the function f (x ) = ln (x ) is the natural logarithm. It is the inverse function to the exponential function g (x ) =

  ex

  where e = 2.7183 is the Euler constant. The inverse has the effect that f (g (x )) = ln(e x ) = x , that is, ln and e cancel each other out.

  A function f (x ) is discontinuous if we have to jump when we move along the graph of the function. For example, consider the graph in Figure 2 . Approaching x = 1 from the left, we have to jump from f (x ) = 1 to f (1) = 0. Thus, the function f is discontinuous at x = 1. Here, f is given by f (x ) = x 2 for x ∈ [0,1), and f (x ) = ln(x ) for x ∈ [1,2).

  Figure 3 Continuity Criterion

  Note : Function f = sin(x ), for −1 ≤ x ≤ 1.

  Formal Derivation

  For a formal treatment of continuity, we first concentrate on the behavior of f at a particular value x* .

  We say that that a function f (x ) is continuous at x * if, for any positive distance δ, we obtain a related distance ε (δ) such that

  What does that mean? We use Figure 3 to illustrate. (The function is f (x ) = sin(x ) with x * = 0.2.) At x *, we have the value f (x *). Now, we select a neighborhood around f (x *) of some arbitrary distance δ as indicated by the dashed horizontal lines through f (x *) − δ and f (x *) + δ, respectively. From the intersections of these horizontal lines and the function graph (solid line), we extend two vertical dash-dotted lines down to the x -axis so that we obtain the two values

  xL

  and

  xU

  , respectively. Now, we measure the distance between

  xL

  and x * and also the distance between

  xU

  and x *. The smaller of the two yields the distance ε (δ). With this distance ε (δ) on the x -axis, we obtain the environment (x * − ε (δ), x * + ε (δ)) about x *. (Note that

  xL = x

  * − ε δ , since the distance between

  xL

  and x * is the shorter one.) The environment is indicated by the dashed lines extending vertically above x * − ε (δ) and x * + ε (δ), respectively. We require that all x that lie in (x * − ε (δ), x * + ε (δ)) yield values f (x ) inside of the environment [f (x *)−δ, f (x *) + δ]. We can see by Figure 3

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  eBook - ePub

  AP® Calculus AB/BC All Access Book + Online + Mobile

  • Stu Schwartz(Author)
  • 2013(Publication Date)
  • Research & Education Association

    (Publisher)

  DIDYOU KNOW

  The sine and cosine functions are fundamental to the theory of periodic functions as those that describe sound and light waves.

  Continuity

  Overview: A very loose definition of a continuous curve is one that can be drawn without picking up the pencil from the paper. Lines are continuous, parabolas are continuous, and so are sine curves. But for more complex curves, we need a definition that can prove where a function is continuous.

  A function f (x) is continuous at x = c if all three conditions hold:

  What this says is that the limit must exist at x = c, the function must have a value at x = c, and that this limit and value must be the same.

  If a function is continuous at all values c in its domain, the function is continuous.

   EXAMPLE 24:

  For each of the following functions, examine their graphs and determine if the function is continuous at the given value of x and, if not, which of the rules of continuity above it fails.

   SOLUTIONS:

  a.  is not continuous at x = 2 because does not exist.

  b.  is not continuous at x = 3 because f (3) does not exist.

  c.  is not continuous at x = 0 because

  d.  is continuous at x = –3.

  e.  is not continuous at x = 1 because does not exist and f (1) does not exist.

  f.  While this is not a continuous curve, it is continuous at x

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  Our Knowledge of the External World

  • Bertrand Russell(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  The application to actual space and time will not be in question to begin with. I do not see any reason to suppose that the points and instants which mathematicians introduce in dealing with space and time are actual physically existing entities, but I do see reason to suppose that the continuity of actual space and time may be more or less analogous to mathematical continuity. The theory of mathematical continuity is an abstract logical theory, not dependent for its validity upon any properties of actual space and time. What is claimed for it is that, when it is understood, certain characteristics of space and time, previously very hard to analyse, are found not to present any logical difficulty. What we know empirically about space and time is insufficient to enable us to decide between various mathematically possible alternatives, but these alternatives are all fully intelligible and fully adequate to the observed facts. For the present, however, it will be well to forget space and time and the continuity of sensible change, in order to return to these topics equipped with the weapons provided by the abstract theory of continuity. Continuity, in mathematics, is a property only possible to a series of terms, i.e. to terms arranged in an order, so that we can say of any two that one comes before the other. Numbers in order of magnitude, the points on a line from left to right, the moments of time from earlier to later, are instances of series. The notion of order, which is here introduced, is one which is not required in the theory of cardinal number. It is possible to know that two classes have the same number of terms without knowing any order in which they are to be taken. We have an instance of this in such a case as English husbands and English wives: we can see that there must be the same number of husbands as of wives, without having to arrange them in a series

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  Deleuze and the History of Mathematics

  In Defense of the 'New'

  • Simon Duffy(Author)
  • 2013(Publication Date)
  • Bloomsbury Academic

    (Publisher)

  The kinds of problems in the infinitesimal calculus that are of interest to Deleuze are those in which the differential relation is generated by differentials and the power series are generated in a process involving the repeated differentiation of the differential relation. In these kinds of problem, it is due to these processes that a function is generated in the first place. The mathematical elements of this interpretation are most clearly developed by Weierstrassian analysis, according to the theorem of the approximation of analytic functions. An analytic function, being secondary to the differential relation, is differentiable, and therefore continuous, at each point of its domain. According to Weierstrass, for any continuous analytic function on a given interval, or domain, there exists a power series expansion which uniformly converges to this function on the given domain. Given that a power series approximates a function in such a restricted domain, the task is then to determine other power series expansions that approximate the same function in other domains. An analytic function is differentiable at each point of its domain, and is essentially defined for Weierstrass from the neighborhood of a singular point by a power series expansion which is convergent with a “circle of convergence” around that point. A power series expansion that is convergent in such a circle represents a function that is analytic at each point in the circle. By taking a point interior to the first circle as a new center, and by determining the values of the coefficients of this new series using the function generated by the first series, a new series and a new center of convergence are obtained, whose circle of convergence overlaps the first. The new series is continuous with the first if the values of the function coincide in the common part of the two circles. This method of “analytic continuity” allows the gradual construction of a whole domain over which the generated function is continuous. At the points of the new circle of convergence that are exterior to, or extend outside, the first, the function represented by the new series is the analytic continuation of the function defined by the first series, what Weierstrass defines as the analytic continuation of a power series expansion outside its circle of convergence. The domain of the function is extended by the successive adjunction of more and more circles of convergence. Each series expansion which determines a circle of convergence is called “an element of the function”. In this way, given an element of an analytic function, by analytic continuation one can obtain the entire analytic function over an extended domain. The domain of the successive adjunction of circles of convergence, as determined by analytic continuity, actually has the structure of a surface. The analytic continuation of power series expansions can be continued in this way in all directions up to the points in the immediate neighborhood exterior to the circles of convergence where the series obtained diverge.

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