Sign function

The sign functions is some sort of a flag that shows what sign a given number has.

Definition.
The sign function is defined for real numbers by

The graph:

Some people use another notation, sgn(x).

One particular application of the sign function is problems with absolute value, since we can write |x| = x⋅sign(x), and more generally, | f (x)| = f (x)⋅sign( f (x)). Here we can use the following fact.

Fact.
If f,g are functions and f is differentiable, then we have

[f⋅sign(g(x))]′ = f ′⋅sign(g(x))

at points x such that g is not zero on some neighborhood of x.

In particular, we can write |x|' = sign(x) for non-zero x and thus write easily derivatives of expressions that feature absolute value. However, it might be misleading since it sort of hides the fact that there is a problem at 0. Thus if you decide to use it, be extra careful.

Sometimes we do not even have to worry. In some settings, the value of functions at individual points do not matter (like in integral transformations). Then we can simply use sign(x) for |x|' without any complications.

Identity function
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