Question

Is `n!` differentiable?

Answer 1

See explanation below.

Explanation:

Since `n!` is a discrete function, only defined for `n>=0 , n inZZ` it is not differentiable as it stands.

However, there is an extension of `n!` into the real numbers;

`f(x) = Gamma(x+1): x in RR` which is continuous and differentiable for `x>=1` and equivalent to `n!` for `x=n in NN`.

See also: http://www.intmath.com/blog/mathematics/factorials-and-the-gamma-function-4350

Hope this makes sense.