The function formula_108 – it oscillates increasingly rapidly between formula_109 and formula_110 as "x" approaches 0. Consequently, the function formula_111 oscillates increasingly rapidly between 0 and formula_112 as "x" approaches 0. If one extends this function by defining formula_113 then the extended function is continuous and everywhere differentiable (it is differentiable at 0 with derivative 0), but has rather unexpected behavior near 0: in any neighborhood of 0 it attains 0 infinitely many times, but also equals formula_112 (a positive number) infinitely often.