linguistic:paraphrasing
Discontinuity at zero

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

These observations are confirmed by any of the various equivalent formal definitions of continuity in mathematical analysis. A function f (x) {\displaystyle f(x)}, such as sgn (x), {\displaystyle \operatorname {sgn}(x),} is continuous at a point x = a {\displaystyle x=a} if the value f (a) {\displaystyle f(a)} can be approximated arbitrarily closely by the sequence of values f (a 1), f (a 2), f (a 3), …, {\displaystyle f(a_{1}),f(a_{2}),f(a_{3}),\dots,} where the a n {\displaystyle a_{n}} make up any infinite sequence which becomes arbitrarily close to a {\displaystyle a} as n {\displaystyle n} becomes sufficiently large. In the notation of mathematical limits, continuity of f {\displaystyle f} at a {\displaystyle a} requires that f (a n) → f (a) {\displaystyle f(a_{n})\to f(a)} as n → ∞ {\displaystyle n\to \infty } for any sequence (a n) n = 1 ∞ {\displaystyle \left(a_{n}\right)_{n=1}^{\infty }} for which a n → a. {\displaystyle a_{n}\to a.} 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 a = 0 {\displaystyle a=0}. For example, we can choose a n {\displaystyle a_{n}} to be the sequence 1, 1 2, 1 3, 1 4, …, {\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots,} which tends towards zero as n {\displaystyle n} increases towards infinity. In this case, a n → a {\displaystyle a_{n}\to a} as required, but sgn (a) = 0 {\displaystyle \operatorname {sgn}(a)=0} and sgn (a n) = + 1 {\displaystyle \operatorname {sgn}(a_{n})=+1} for each n, {\displaystyle n,} so that sgn (a n) → 1 ≠ sgn (a) {\displaystyle \operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)}. This counterexample confirms more formally the discontinuity of sgn x {\displaystyle \operatorname {sgn} x} 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.