Editing Sign function (section)

=== Discontinuity at zero ===
[[File:Discontinuity of the sign function at 0.svg|thumb|300px|The sign function is not [[continuous function | continuous]] at <math>x=0</math>.]]

Although the sign function takes the value {{math|&minus;1}} when <math>x</math> is negative, the ringed point {{math|(0, &minus;1)}} in the plot of <math>\sgn x</math> indicates that this is not the case when <math>x=0</math>. Instead, the value jumps abruptly to the solid point at {{math|(0, 0)}} where <math>\sgn(0)=0</math>. There is then a similar jump to <math>\sgn(x)=+1</math> when <math>x</math> is positive. Either jump demonstrates visually that the sign function <math>\sgn x</math> is discontinuous at zero, even though it is continuous at any point where <math>x</math> is either positive or negative.

These observations are confirmed by any of the various equivalent formal definitions of [[Continuous function|continuity]] in [[mathematical analysis]]. A function <math>f(x)</math>, such as <math>\sgn(x),</math> is continuous at a point <math>x=a</math> if the value <math>f(a)</math> can be approximated arbitrarily closely by the [[sequence]] of values <math>f(a_1),f(a_2),f(a_3),\dots,</math> where the <math>a_n</math> make up any infinite sequence which becomes arbitrarily close to <math>a</math> as <math>n</math> becomes sufficiently large. In the notation of mathematical [[Limit of a sequence|limit]]s, continuity of <math>f</math> at <math>a</math> requires that <math>f(a_n) \to f(a)</math> as <math>n \to \infty</math> for any sequence <math>\left(a_n\right)_{n=1}^\infty</math> for which <math>a_n \to a.</math> The arrow symbol can be read to mean ''approaches'', or ''tends to'', and it applies to the sequence as a whole.

This criterion fails for the sign function at <math>a=0</math>. For example, we can choose <math>a_n</math> to be the sequence <math>1,\tfrac{1}{2},\tfrac{1}{3},\tfrac{1}{4},\dots,</math> which tends towards zero as <math>n</math> increases towards infinity. In this case, <math>a_n \to a</math> as required, but <math>\sgn(a)=0</math> and <math>\sgn(a_n)=+1</math> for each <math>n,</math> so that <math>\sgn(a_n) \to 1 \neq \sgn(a)</math>. This counterexample confirms more formally the discontinuity of <math>\sgn x</math> at zero that is visible in the plot.

Despite the sign function having a very simple form, the step change at zero causes difficulties for traditional [[calculus]] techniques, which are quite stringent in their requirements. Continuity is a frequent constraint. One solution can be to approximate the sign function by a smooth continuous function; others might involve less stringent approaches that build on classical methods to accommodate larger classes of function.