471DiscontinuousFunctions.nb

471: Discontinuities of the First and Second Kind

•Definitions

•Definition: (right-hand and left-hand limits, Rudin's definition)

Let f be defined on (a,b) ⊂ . For a <= x < b, f(x+) = q if and only if as , for all sequences ${t _ n}$ in (x.b) such that . We call f(x+) the right-hand limit of f at x. Similarly, for a < x <= b, f(x-) = q if and only if as , for all sequences ${t _ n}$ in (a,x) such that . We call f(x-) the left-hand limit of f at x.

•Definition: (right-hand and left-hand limits, ∈-δ definition)

Let f : (a,b) ⊂ -> Y, where Y is a metric space. The right-hand limit of f at x=a is defined as follows. if and only if given ∈ > 0, there exists a δ > 0 such that , whenever a < x < a + δ. The left-hand limit of f at x=b is defined as follows. if and only if given ∈ > 0, there exists a δ > 0 such that , whenever b - δ < x < b.

•Note: exists if and only if = .

•Definition: (types of discontinuities)

Let f be defined on (a,b) ⊂ , except possibly at x ∈ (a,b). If f is discontinuous at x and f(x+), f(x-) exist, then we say f has a simple discontinuity (discontinuity of the first kind) at x. Otherwise, we say f has a discontinuity of the second kind) at x.

•Example 1:

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f(x+) and f(x-) do not exist for any x ∈ , since for any sequence of rational numbers ${t _ n}$ such that , and for any sequence of irrational numbers ${t _ n}$ such that , .·. f has a discontinuity of the second kind at any real number x.

•Example 2:

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f(x+) and f(x-) do not exist for any x ∈ \{0}, since for any sequence of rational numbers ${t _ n}$ such that , and for any sequence of irrational numbers ${t _ n}$ such that , .·. f has a discontinuity of the second kind at any real number x != 0. Note that f is continuous at x = 0.

•Example 3:

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f is continuous at any x in the interval (-3,1), except at x=0. f(0+) =2 and f(0-) = -2, so f has a discontinuity of the first kind at x = 0.

•Example 4:

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f(0+) is undefined, since as , for all n ∈ , on this function takes on all values of one period of the sine function. Similarly, f(0-) is undefined. .·. f has a discontinuity of the second kind at x = 0. For all other x - values, this function is continuous. Note that f is not uniformly continuous on (0,∞), as given , there exists an integer such that ⊂ (0,δ).

•Example 5:

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Since f(0+) = f(0-) = 1, this function is continuous at x = 0. f is continuous at all x - values. Note that without the definition of this function being 1 at x = 0, f would have a discontinuity of the first kind at x = 0. (We say f(x) = has a removable discontinuity at x = 0.)

•Example 6:

$Plot[(x - 1) * Abs[x - 2]/(x^2 - 3 x + 2), {x, -2, 3}, Frame -> True] ;$

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f has a removable discontinuity at x = 1. The simple discontinuity at x = 2 cannot be removed.

•Example 7: $f(x) = { 1 x, if x = -, n ∈ їН \ {0} n 0, otherwise$

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f has a simple discontinuity at x = for any n ∈ Z\{0}, since . Note that f is continuous at x = 0, even though you need to "pick up your pencil" arbitrarily close to the origin in order to draw the graph. This is an example where the intuitive idea of continuity would fail to show f is continuous at x = 0!!

•Example 8:

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$Plot[1/x^2, {x, -2, 2}, Frame -> True, PlotRange -> {-50, 50}] ;$

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Each of these functions is continuous at all x - values except at x = 0. At x = 0 these functions have a discontinuity of the second kind. (In Calculus I, we'd say that as , , as , , as ,, and as . See Rudin, pp. 97 - 98 for a discussion of infinite limits and limits at infinity.)

Converted by Mathematica (December 1, 2003)