A piecewise function can definitely be differentiable if (a) its pieces are differentiable and (b) it's differentiable at the points where they're joined. For example, if f(x) = 0 for x <= 0 and 1 for x > 0, (a) is true because the pieces are differentiable, but b is not because it's not differentiable at x = 0.

Is piecewise continuously differentiable?

A piecewise continuously differentiable function is referred to in some sources as a piecewise smooth function. However, as a smooth function is defined on Pr∞fWiki as being of differentiability class ∞, this can cause confusion, so is not recommended.

What is piecewise differentiable?

Given any real numbers α and β such that α < β, a continuous function f : [α, β] → R is said to be piecewise differentiable if there exist n ∈ N \ {0} and points x1, ..., xn in [α, β] such that x0 = α<x1 < ··· < xn < β = xn+1 and for any i ∈ {0,...,n}, the restriction of f on (xi,xi+1) is everywhere differentiable.