Stewart – Calculus – 2.5 – Continuity

  1. Definition: A function f is continuous at a number a if

    This definition implicitly requires three things if f is continuous at a:

    1. f(a) is defined (that is, a is in the domain of f)

    2. \lim_{x \to a}f(x) exists

    3. \lim_{x \to a}f(x)=f(a)

  2. If f and g are continuous functions with f(3)=5 and
    \lim_{x \to 3}[2f(x) - g(x)]=4, find g(3).

    Use the definition of continuity and the properties of limits to show that the funtion is continous at the given number a. f(x)=(x+2x^{3})^{4}, a=-1

  3. The Intermediate Value Theorem
    Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) \neq f(b). Then there exists a number c in (a,b) such that f(c)=N.

    ~ ~ ~

    Use the Intermediate Value Theorem to show that there is
    a root of the given equation in the specified interval.

    x^4+x-3=0, (1,2)

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